The fmincon function finds the minimum of a constrained nonlinear multivariable function subject to bounds and linear and nonlinear constraints. It uses either a medium-scale or large-scale algorithm depending on whether the gradient of the objective function is provided. The user defines the objective function fun and optional nonlinear constraints nonlcon, both of which can return gradient and Hessian information. Fmincon returns the optimized values, objective function value, exit condition, and additional output details.
This document provides an overview of Markov Decision Processes (MDPs) and related concepts in decision theory and reinforcement learning. It defines MDPs and their components, describes algorithms for solving MDPs like value iteration and policy iteration, and discusses extensions to partially observable MDPs. It also briefly mentions dynamic Bayesian networks, the dopaminergic system, and its role in reinforcement learning and decision making.
The document provides an overview of the Laplace transform:
1. It introduces the Laplace transform and describes how it is used to transform functions from the time domain to the complex s-domain. This allows solving circuit problems involving initial conditions using algebraic equations rather than differential equations.
2. Key properties and theorems of the Laplace transform are described, including its use in solving linear time-invariant differential equations by taking the Laplace transform of both sides of the equation.
3. The inverse Laplace transform is explained as a way to transform signals back from the s-domain to the time domain. Common Laplace transform pairs and the Laplace transforms of basic circuit elements are also summarized.
The document discusses problem-solving agents and search algorithms. It provides examples of toy problems like the 8-puzzle and real-world problems like touring in Romania. Problem-solving agents work by formulating a goal, formulating the problem as a set of states and actions, and then using a search algorithm to find a solution. Real-world problems are more complex to define than toy problems and people care about the solutions. The document also provides examples of defining the state space, actions, goal tests, and path costs for various problems.
The document describes pushdown automata (PDA) which are analogous to context-free languages in the same way that finite automata are analogous to regular languages. A PDA has states, input symbols, stack symbols, transition functions, an initial state, initial stack symbol, and accepting states. The transition function specifies state transitions based on the current state, input symbol, and top of stack symbol and can modify the stack. The document provides examples of PDAs for languages of the form wwr and balanced parentheses and discusses how PDAs work by changing their instantaneous descriptions as the input is processed and stack is modified.
This document discusses using fuzzy multi-objective linear programming (FMOLP) to solve the traveling salesman problem (TSP). TSP aims to find the shortest route to visit each city once. FMOLP handles fuzzy constraints and goals. It formulates TSP as three objectives - minimizing cost, distance, and time. A case study applies FMOLP to find the optimal route between 4 cities. The solution maximizes satisfaction level α while meeting fuzzy constraints on each objective.
This document outlines an introduction to convex optimization. It begins with an introduction stating that convex optimization problems can be solved efficiently to find the global optimum. It then provides an outline covering convex sets, convex functions, convex optimization problems, and references. The body of the document defines convex sets as sets where a line segment between any two points lies entirely within the set. It also provides examples of convex sets including norm balls and intersections of convex sets. It defines convex functions as functions where the graph lies below any line segment between two points, and provides conditions for checking convexity using derivatives. Finally, it discusses convex optimization problems and solving them efficiently.
This document provides an overview of Markov Decision Processes (MDPs) and related concepts in decision theory and reinforcement learning. It defines MDPs and their components, describes algorithms for solving MDPs like value iteration and policy iteration, and discusses extensions to partially observable MDPs. It also briefly mentions dynamic Bayesian networks, the dopaminergic system, and its role in reinforcement learning and decision making.
The document provides an overview of the Laplace transform:
1. It introduces the Laplace transform and describes how it is used to transform functions from the time domain to the complex s-domain. This allows solving circuit problems involving initial conditions using algebraic equations rather than differential equations.
2. Key properties and theorems of the Laplace transform are described, including its use in solving linear time-invariant differential equations by taking the Laplace transform of both sides of the equation.
3. The inverse Laplace transform is explained as a way to transform signals back from the s-domain to the time domain. Common Laplace transform pairs and the Laplace transforms of basic circuit elements are also summarized.
The document discusses problem-solving agents and search algorithms. It provides examples of toy problems like the 8-puzzle and real-world problems like touring in Romania. Problem-solving agents work by formulating a goal, formulating the problem as a set of states and actions, and then using a search algorithm to find a solution. Real-world problems are more complex to define than toy problems and people care about the solutions. The document also provides examples of defining the state space, actions, goal tests, and path costs for various problems.
The document describes pushdown automata (PDA) which are analogous to context-free languages in the same way that finite automata are analogous to regular languages. A PDA has states, input symbols, stack symbols, transition functions, an initial state, initial stack symbol, and accepting states. The transition function specifies state transitions based on the current state, input symbol, and top of stack symbol and can modify the stack. The document provides examples of PDAs for languages of the form wwr and balanced parentheses and discusses how PDAs work by changing their instantaneous descriptions as the input is processed and stack is modified.
This document discusses using fuzzy multi-objective linear programming (FMOLP) to solve the traveling salesman problem (TSP). TSP aims to find the shortest route to visit each city once. FMOLP handles fuzzy constraints and goals. It formulates TSP as three objectives - minimizing cost, distance, and time. A case study applies FMOLP to find the optimal route between 4 cities. The solution maximizes satisfaction level α while meeting fuzzy constraints on each objective.
This document outlines an introduction to convex optimization. It begins with an introduction stating that convex optimization problems can be solved efficiently to find the global optimum. It then provides an outline covering convex sets, convex functions, convex optimization problems, and references. The body of the document defines convex sets as sets where a line segment between any two points lies entirely within the set. It also provides examples of convex sets including norm balls and intersections of convex sets. It defines convex functions as functions where the graph lies below any line segment between two points, and provides conditions for checking convexity using derivatives. Finally, it discusses convex optimization problems and solving them efficiently.
This presentation discusses context-free grammars. It defines context-free grammars and provides an example. It also discusses parse trees, including how they are generated and different types (top-down and bottom-up). Examples are provided to demonstrate leftmost and rightmost derivations and parse trees. The document concludes that the grammar presented, with production rules of X → X+X | X*X |X| a, is ambiguous as there are two possible parse trees for the string "a+a*a".
The document contains questions and answers related to compiler design topics such as parsing, grammars, syntax analysis, error handling, derivation, sentential forms, parse trees, ambiguity, left and right recursion elimination etc. Key points discussed are:
1. The role of parser is to verify the string of tokens generated by lexical analyzer according to the grammar rules and detect syntax errors. It outputs a parse tree.
2. Common parsing methods are top-down, bottom-up and universal. Top-down methods include LL, LR. Bottom-up methods include LR, LALR.
3. Errors can be lexical, syntactic, semantic and logical detected by different compiler phases. Error recovery strategies include panic mode
The document discusses numerical methods for solving ordinary differential equations (ODEs), including Taylor's series method and Picard's method. It provides examples of applying Taylor's series method to approximate solutions of first order ODEs at different values of x to 4-5 decimal places of accuracy. The examples given include solving ODEs with initial conditions and computing solutions at multiple x values by taking terms from the Taylor series expansion.
This document summarizes integer programming and two methods for solving integer programming problems: Gomory's fractional cut method and branch and bound method.
Gomory's fractional cut method involves first solving the integer programming problem as a linear programming problem by ignoring integer restrictions. If the solution is non-integer, a new constraint called a fractional cut is generated and added to obtain an integer solution. This process repeats until an optimal integer solution is found.
Branch and bound is a search method that involves constructing a tree by branching on integer variables. It prunes branches that cannot produce better solutions than the best known solution to reduce computation.
The document provides an example to illustrate Gomory's fractional cut method and its
Performance analysis of a second order system using mraciaemedu
The document analyzes the performance of a second-order system using model reference adaptive control (MRAC) with two different adaptation rules - MIT rule and Lyapunov rule. It first describes the MRAC scheme and the two adaptation rules. It then models a second-order underdamped plant and a critically damped reference model. Lastly, it applies the MIT rule to the system and derives the adaptation laws to minimize the error between the plant and reference model outputs. The analysis aims to improve the dynamic performance of the underdamped plant using MRAC.
The document discusses feedback linearization, which involves transforming a nonlinear system into an equivalent linear system through state and input transformations. This allows linear control techniques to be applied. Feedback linearization has been successfully used to control helicopters, aircraft, robots, and biomedical devices. It involves finding transformations to cancel the system nonlinearities, resulting in a linear relationship between the transformed input and states. The transformed system can then be stabilized using standard linear control methods like pole placement.
This document provides an overview of optimization techniques. It begins with an introduction to optimization and examples of optimization problems. It then discusses the historical development of optimization methods. The document categorizes optimization problems into convex, concave, and subclasses like linear programming, quadratic programming, and semidefinite programming. It also covers advanced optimization methods like interior point methods. Finally, the document discusses applications of convex optimization techniques in fields such as engineering, finance, and wireless communications.
The document presents a logic puzzle about a monkey trying to reach bananas hung from the ceiling. It describes the problem setup, with a monkey and bananas in a room, along with a chair. It then provides the logical representation of the problem using predicates and axioms. The goal is to use resolution logic to prove that the monkey can reach the bananas by placing the chair below them and climbing on top of it. Through a series of resolvents applying the given information, it is shown that the monkey can indeed reach the bananas.
The document discusses the grey wolf optimizer (GWO) algorithm, which is a meta-heuristic algorithm inspired by grey wolves' hunting behavior. It describes the social hierarchy of grey wolves, including alpha, beta, delta, and omega ranks. The algorithm simulates grey wolves' hunting techniques like encircling prey, hunting guided by the alpha/beta/delta ranks, attacking prey through exploitation, and searching for prey through exploration. The GWO algorithm initializes parameters and a population, assigns the best three solutions, updates other solutions, and iterates until termination criteria are met to find the best solution.
Optimization involves finding the best values for variables to minimize or maximize an objective function subject to constraints. An optimization problem consists of an objective function, variables, and constraints. The objective function expresses the performance of a system and must be minimized or maximized. Variables define the objective function and constraints. Constraints allow variables to take on certain values but exclude others to ensure feasibility. Common optimization techniques include mathematical programming, calculus methods, network methods, and meta-heuristic algorithms such as genetic algorithms, simulated annealing, and whale optimization.
This document discusses Mealy and Moore machines, which are two types of finite state machines (FSM). It provides details on each:
- Mealy machines produce outputs on transitions between states based on both the present state and input. Moore machines produce outputs based only on the present state.
- Both can be described using tuples but the output functions differ - Mealy machines have an output transition function while Moore machines have an output function.
- Advantages of each include Mealy machines having faster response times while Moore machines are cheaper and easier to use. Disadvantages include Mealy machines being more expensive and states becoming unmanageable.
- Examples of uses include processors for Mealy machines and SRAM
Existence, Uniqueness and Stability Solution of Differential Equations with B...iosrjce
In this work, we investigate the existence ,uniqueness and stability solution of non-linear
differential equations with boundary conditions by using both method Picard approximation and
Banach fixed point theorem which were introduced by [6] .These investigations lead us to improving
and extending the above method. Also we expand the results obtained by [1] to change the non-linear
differential equations with initial condition to non-linear differential equations with boundary
conditions
An adaptative nature inspired algorithm explained, concretely implemented, and applied to routing protocols in wired and wireless networks. The document discusses how ant colony optimization algorithms can be applied to routing by simulating how ants leave pheromone trails to find the shortest path between their nest and food sources. It provides examples of how ant colony algorithms have been implemented in routing protocols like ABC for wired networks, AntNet for MANETs, and ARA and AntHocNet for wireless ad hoc networks. Evaluation results show these ant-inspired routing protocols can find paths more efficiently than traditional routing protocols like OSPF and perform better than protocols like AODV for packet delivery in mobile wireless networks.
The document describes the steps of the dual simplex method for solving a maximization linear programming problem. It begins with ensuring all reduced costs in the simplex tableau are nonnegative before attempting the method. The key steps are: (1) check if the right-hand sides are nonnegative and stop if so, (2) pick an exiting variable if a right-hand side is negative, (3) use the minimum ratio test to select an entering variable, and (4) pivot and return to step 1. The example problem demonstrates applying these steps to solve a maximization problem using the dual simplex method.
This document discusses optimization problems and their solutions. It begins by defining optimization problems as seeking to maximize or minimize a quantity given certain limits or constraints. Both deterministic and stochastic models are discussed. Examples of discrete optimization problems include the traveling salesman and shortest path problems. Solution methods mentioned include integer programming, network algorithms, dynamic programming, and approximation algorithms. The document then focuses on convex optimization problems, which can be solved efficiently. It discusses using tools like CVX for solving convex programs and the duality between primal and dual problems. Finally, it presents the collaborative resource allocation algorithm for solving non-convex optimization problems in a suboptimal way.
Design and analysis of robust h infinity controllerAlexander Decker
1) The document discusses the design and analysis of an H-infinity controller. H-infinity control guarantees robustness and good performance through high disturbance rejection.
2) It presents a simplified step-by-step procedure for designing an H-infinity controller for a given system using H-infinity loop shaping. This technique allows performance requirements to be incorporated into the design through the use of performance weights.
3) The generalized plant model is augmented with weight functions to shape the closed-loop response to meet design specifications such as uncertainty attenuation and required bandwidth. The H-infinity controller is then synthesized to minimize sensitivity and complementarity weights.
Quadratic programming (Tool of optimization)VARUN KUMAR
This document provides an overview of quadratic programming, including:
1. It defines quadratic programming as a special case of nonlinear programming where the objective function is quadratic and all constraints are linear.
2. It presents the general mathematical formulation of a quadratic programming problem and provides an example problem.
3. It discusses solutions to quadratic programs using the graphical method and the Karush-Kuhn-Tucker (KKT) conditions.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
This document contains a summary of key concepts in mathematics functions including:
- Definitions of different types of functions such as polynomial, rational, trigonometric, exponential, and logarithmic functions.
- Concepts related to functions like domain, range, and notation for functions.
- Examples demonstrating how to find the domain and range of functions.
- Key topics in trigonometry including trigonometric functions, trigonometric tables, addition formulas, laws of sines and cosines.
- Concepts of limits, average and instantaneous rates of change, techniques for finding limits, and limits involving infinity.
This document discusses concepts related to calculus including functions, limits, continuity, and derivatives. The objective is for students to be able to evaluate limits and determine derivatives of algebraic functions. It defines functions and function notation. It discusses limits, continuity, and the definition of the derivative. It provides examples of evaluating limits using theorems and the squeeze principle. It also defines types of discontinuities and conditions for continuity.
This presentation discusses context-free grammars. It defines context-free grammars and provides an example. It also discusses parse trees, including how they are generated and different types (top-down and bottom-up). Examples are provided to demonstrate leftmost and rightmost derivations and parse trees. The document concludes that the grammar presented, with production rules of X → X+X | X*X |X| a, is ambiguous as there are two possible parse trees for the string "a+a*a".
The document contains questions and answers related to compiler design topics such as parsing, grammars, syntax analysis, error handling, derivation, sentential forms, parse trees, ambiguity, left and right recursion elimination etc. Key points discussed are:
1. The role of parser is to verify the string of tokens generated by lexical analyzer according to the grammar rules and detect syntax errors. It outputs a parse tree.
2. Common parsing methods are top-down, bottom-up and universal. Top-down methods include LL, LR. Bottom-up methods include LR, LALR.
3. Errors can be lexical, syntactic, semantic and logical detected by different compiler phases. Error recovery strategies include panic mode
The document discusses numerical methods for solving ordinary differential equations (ODEs), including Taylor's series method and Picard's method. It provides examples of applying Taylor's series method to approximate solutions of first order ODEs at different values of x to 4-5 decimal places of accuracy. The examples given include solving ODEs with initial conditions and computing solutions at multiple x values by taking terms from the Taylor series expansion.
This document summarizes integer programming and two methods for solving integer programming problems: Gomory's fractional cut method and branch and bound method.
Gomory's fractional cut method involves first solving the integer programming problem as a linear programming problem by ignoring integer restrictions. If the solution is non-integer, a new constraint called a fractional cut is generated and added to obtain an integer solution. This process repeats until an optimal integer solution is found.
Branch and bound is a search method that involves constructing a tree by branching on integer variables. It prunes branches that cannot produce better solutions than the best known solution to reduce computation.
The document provides an example to illustrate Gomory's fractional cut method and its
Performance analysis of a second order system using mraciaemedu
The document analyzes the performance of a second-order system using model reference adaptive control (MRAC) with two different adaptation rules - MIT rule and Lyapunov rule. It first describes the MRAC scheme and the two adaptation rules. It then models a second-order underdamped plant and a critically damped reference model. Lastly, it applies the MIT rule to the system and derives the adaptation laws to minimize the error between the plant and reference model outputs. The analysis aims to improve the dynamic performance of the underdamped plant using MRAC.
The document discusses feedback linearization, which involves transforming a nonlinear system into an equivalent linear system through state and input transformations. This allows linear control techniques to be applied. Feedback linearization has been successfully used to control helicopters, aircraft, robots, and biomedical devices. It involves finding transformations to cancel the system nonlinearities, resulting in a linear relationship between the transformed input and states. The transformed system can then be stabilized using standard linear control methods like pole placement.
This document provides an overview of optimization techniques. It begins with an introduction to optimization and examples of optimization problems. It then discusses the historical development of optimization methods. The document categorizes optimization problems into convex, concave, and subclasses like linear programming, quadratic programming, and semidefinite programming. It also covers advanced optimization methods like interior point methods. Finally, the document discusses applications of convex optimization techniques in fields such as engineering, finance, and wireless communications.
The document presents a logic puzzle about a monkey trying to reach bananas hung from the ceiling. It describes the problem setup, with a monkey and bananas in a room, along with a chair. It then provides the logical representation of the problem using predicates and axioms. The goal is to use resolution logic to prove that the monkey can reach the bananas by placing the chair below them and climbing on top of it. Through a series of resolvents applying the given information, it is shown that the monkey can indeed reach the bananas.
The document discusses the grey wolf optimizer (GWO) algorithm, which is a meta-heuristic algorithm inspired by grey wolves' hunting behavior. It describes the social hierarchy of grey wolves, including alpha, beta, delta, and omega ranks. The algorithm simulates grey wolves' hunting techniques like encircling prey, hunting guided by the alpha/beta/delta ranks, attacking prey through exploitation, and searching for prey through exploration. The GWO algorithm initializes parameters and a population, assigns the best three solutions, updates other solutions, and iterates until termination criteria are met to find the best solution.
Optimization involves finding the best values for variables to minimize or maximize an objective function subject to constraints. An optimization problem consists of an objective function, variables, and constraints. The objective function expresses the performance of a system and must be minimized or maximized. Variables define the objective function and constraints. Constraints allow variables to take on certain values but exclude others to ensure feasibility. Common optimization techniques include mathematical programming, calculus methods, network methods, and meta-heuristic algorithms such as genetic algorithms, simulated annealing, and whale optimization.
This document discusses Mealy and Moore machines, which are two types of finite state machines (FSM). It provides details on each:
- Mealy machines produce outputs on transitions between states based on both the present state and input. Moore machines produce outputs based only on the present state.
- Both can be described using tuples but the output functions differ - Mealy machines have an output transition function while Moore machines have an output function.
- Advantages of each include Mealy machines having faster response times while Moore machines are cheaper and easier to use. Disadvantages include Mealy machines being more expensive and states becoming unmanageable.
- Examples of uses include processors for Mealy machines and SRAM
Existence, Uniqueness and Stability Solution of Differential Equations with B...iosrjce
In this work, we investigate the existence ,uniqueness and stability solution of non-linear
differential equations with boundary conditions by using both method Picard approximation and
Banach fixed point theorem which were introduced by [6] .These investigations lead us to improving
and extending the above method. Also we expand the results obtained by [1] to change the non-linear
differential equations with initial condition to non-linear differential equations with boundary
conditions
An adaptative nature inspired algorithm explained, concretely implemented, and applied to routing protocols in wired and wireless networks. The document discusses how ant colony optimization algorithms can be applied to routing by simulating how ants leave pheromone trails to find the shortest path between their nest and food sources. It provides examples of how ant colony algorithms have been implemented in routing protocols like ABC for wired networks, AntNet for MANETs, and ARA and AntHocNet for wireless ad hoc networks. Evaluation results show these ant-inspired routing protocols can find paths more efficiently than traditional routing protocols like OSPF and perform better than protocols like AODV for packet delivery in mobile wireless networks.
The document describes the steps of the dual simplex method for solving a maximization linear programming problem. It begins with ensuring all reduced costs in the simplex tableau are nonnegative before attempting the method. The key steps are: (1) check if the right-hand sides are nonnegative and stop if so, (2) pick an exiting variable if a right-hand side is negative, (3) use the minimum ratio test to select an entering variable, and (4) pivot and return to step 1. The example problem demonstrates applying these steps to solve a maximization problem using the dual simplex method.
This document discusses optimization problems and their solutions. It begins by defining optimization problems as seeking to maximize or minimize a quantity given certain limits or constraints. Both deterministic and stochastic models are discussed. Examples of discrete optimization problems include the traveling salesman and shortest path problems. Solution methods mentioned include integer programming, network algorithms, dynamic programming, and approximation algorithms. The document then focuses on convex optimization problems, which can be solved efficiently. It discusses using tools like CVX for solving convex programs and the duality between primal and dual problems. Finally, it presents the collaborative resource allocation algorithm for solving non-convex optimization problems in a suboptimal way.
Design and analysis of robust h infinity controllerAlexander Decker
1) The document discusses the design and analysis of an H-infinity controller. H-infinity control guarantees robustness and good performance through high disturbance rejection.
2) It presents a simplified step-by-step procedure for designing an H-infinity controller for a given system using H-infinity loop shaping. This technique allows performance requirements to be incorporated into the design through the use of performance weights.
3) The generalized plant model is augmented with weight functions to shape the closed-loop response to meet design specifications such as uncertainty attenuation and required bandwidth. The H-infinity controller is then synthesized to minimize sensitivity and complementarity weights.
Quadratic programming (Tool of optimization)VARUN KUMAR
This document provides an overview of quadratic programming, including:
1. It defines quadratic programming as a special case of nonlinear programming where the objective function is quadratic and all constraints are linear.
2. It presents the general mathematical formulation of a quadratic programming problem and provides an example problem.
3. It discusses solutions to quadratic programs using the graphical method and the Karush-Kuhn-Tucker (KKT) conditions.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
This document contains a summary of key concepts in mathematics functions including:
- Definitions of different types of functions such as polynomial, rational, trigonometric, exponential, and logarithmic functions.
- Concepts related to functions like domain, range, and notation for functions.
- Examples demonstrating how to find the domain and range of functions.
- Key topics in trigonometry including trigonometric functions, trigonometric tables, addition formulas, laws of sines and cosines.
- Concepts of limits, average and instantaneous rates of change, techniques for finding limits, and limits involving infinity.
This document discusses concepts related to calculus including functions, limits, continuity, and derivatives. The objective is for students to be able to evaluate limits and determine derivatives of algebraic functions. It defines functions and function notation. It discusses limits, continuity, and the definition of the derivative. It provides examples of evaluating limits using theorems and the squeeze principle. It also defines types of discontinuities and conditions for continuity.
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The domain is the set of all possible x-values, and the range is the resulting y-values. The rate of change of a function can be found using the difference quotient or slope formula. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point. Domain restrictions limit the possible input values of a function. Horizontal and vertical asymptotes provide information about the behavior of a rational function as x approaches positive or negative infinity.
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The domain is the set of all possible x-values, and the range is the resulting y-values. To find the rate of change, use the difference quotient which is similar to the slope formula. The vertical line test determines if a graph represents a function - if any vertical line intersects the graph more than once, it is not a function. Domain restrictions specify allowed input values.
This document provides information about the key concepts and formulas related to differential calculus. It defines the derivative as the rate of change of a function with respect to an independent variable. Some common derivative formulas are presented for various standard functions like polynomial, exponential, logarithmic and trigonometric functions. Examples are given to demonstrate calculating derivatives using the formulas. The concepts of maxima and minima of functions are also introduced along with an example to find the maximum and minimum values of a given function without using derivatives.
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The rate of change of a function is found by taking the difference quotient, which is similar to the slope formula. The domain of a function is the set of all possible x-values, while the range is the resulting y-values. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point.
This document discusses algorithms and their analysis. It begins by defining an algorithm as a precise set of instructions to solve a problem or perform a computation. Key properties of algorithms are described, including having specified inputs/outputs and being finite, definite, and effective. Examples are given of algorithms to find the maximum element, perform linear search, and perform binary search on an ordered list. The document then discusses analyzing the time and space complexity of algorithms as their input size increases. Common complexity functions like constant, logarithmic, linear, quadratic, and exponential time are described. Rules for analyzing the complexity of combined operations are provided. Finally, examples are worked through to determine the time complexity of algorithms that find the maximum difference between elements in a list
This document provides an overview of key calculus concepts including:
- Functions and function notation which are fundamental to calculus
- Limits which allow defining new points from sequences and are essential to calculus concepts like derivatives and integrals
- Derivatives which measure how one quantity changes in response to changes in another related quantity
- Types of infinity and limits involving infinite quantities or areas
The document defines functions, limits, derivatives, and infinity, and provides examples to illustrate these core calculus topics. It lays the groundwork for further calculus concepts to be covered like integrals, derivatives of more complex functions, and applications of limits, derivatives, and infinity.
This document discusses Python functions. It covers built-in functions, user-defined functions, fruitful and void functions, parameters and arguments, recursion, and strings. Specifically, it provides examples and explanations of various math functions like abs(), ceil(), exp(), and more. It also covers defining new functions, importing functions from modules, boolean functions, and using for loops to traverse strings.
This document discusses various topics related to piecewise functions and rational functions:
- It defines piecewise functions and provides examples of evaluating piecewise functions at given values.
- It introduces rational functions as functions of the form p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not equal to zero. It discusses representing rational functions in different forms.
- It explains how to identify restrictions or extraneous roots of rational functions by setting the denominator equal to zero. It also discusses how to determine the domain of a rational function based on its restrictions.
- Finally, it defines vertical and horizontal asymptotes of rational functions. It provides
A Level Set Method For Multiobjective Combinatorial Optimization Application...Scott Faria
This document proposes a new algorithm for computing all Pareto optimal solutions to multiobjective combinatorial optimization problems based on the level set method. The algorithm generates level sets in order of increasing objective function values for one objective at a time, checking if each solution is contained in the other level sets and dominates previously found solutions. It relies on the ability to find the K best solutions to a single objective combinatorial problem. The method is applied to the multiobjective quadratic assignment problem and computational results are presented.
Matlab is a high-level programming language and environment used for numerical computation, visualization, and programming. The document outlines key Matlab concepts including the Matlab screen, variables, arrays, matrices, operators, plotting, flow control, m-files, and user-defined functions. Matlab allows users to analyze data, develop algorithms, and create models and applications.
This document provides an introduction to using MATLAB. It begins with instructions on starting and exiting MATLAB. It then discusses MATLAB's basic functionality for matrix calculations and operations. Subsequent sections cover topics like repeating commands, subscripting matrices, the edit-test-edit cycle for developing code, writing functions and scripts, and input/output in MATLAB including loading/saving data and printing output. Exercises are provided throughout to help readers practice key concepts.
X01 Supervised learning problem linear regression one feature theorieMarco Moldenhauer
1. The document describes supervised learning problems, specifically linear regression with one feature. It defines key concepts like the hypothesis function, cost function, and gradient descent algorithm.
2. A data set with one input feature and one output is defined. The goal is to learn a linear function that maps the input to the output to best fit the training data.
3. The hypothesis function is defined as h(x) = θ0 + θ1x, where θ0 and θ1 are parameters to be estimated. Gradient descent is used to minimize the cost function and find the optimal θ values.
1) The document outlines a course curriculum that covers functions, rational functions, one-to-one functions, exponential functions, and logarithmic functions over 32 hours spread across 8 weeks.
2) It provides the chapter titles and learning objectives for each chapter, along with the topics and hours allocated to each lesson.
3) Key concepts covered include functions as models of real-life situations, representing functions as sets of ordered pairs, tables, graphs, and piecewise functions.
This presentation is a part of Computer Oriented Numerical Method . Newton-Cotes formulas are an extremely useful and straightforward family of numerical integration techniques.
This document provides an introduction to machine learning concepts including supervised and unsupervised learning. It discusses linear regression with one variable and multiple variables. For linear regression with one variable, it describes the hypothesis function, cost function, gradient descent algorithm, and makes predictions using a housing dataset. For multiple variables, it introduces feature normalization and applies the concepts to predict housing prices based on size, bedrooms and price in a real estate dataset. The document provides code examples to implement the algorithms.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
1) Functions relate inputs to outputs through ordered pairs where each input maps to exactly one output. The domain is the set of inputs and the range is the set of outputs.
2) There are different types of functions including linear, quadratic, and composition functions. A linear function's graph is a straight line while a quadratic function's graph is a parabola.
3) Composition functions combine other functions where the output of one becomes the input of another. Together functions provide a powerful modeling tool used across many fields including medicine.
The document contains critical data for various compounds including their molecular weights, freezing points, boiling points, critical temperatures, pressures, volumes, compressibility factors, and acentric factors. The data is presented in a table with 27 compounds listed along with their properties. This table provides physical property data for analysis of these compounds and calculations involving their critical points and states.
Influence of temperature on the liquid liquid equilibria of methanol benzene ...Josemar Pereira da Silva
This document summarizes a study on the liquid-liquid equilibrium of the ternary system composed of methanol, benzene, and hexane at different temperatures. Experimental data on the compositions of the coexisting liquid phases are reported at 278.15 K, 283.15 K, and 293.15 K. The data show that increasing the temperature decreases the miscibility gap between the methanol-rich and hexane-rich phases. Models like UNIQUAC and NRTL are able to correlate the experimental equilibrium data reasonably well at the different temperatures, while UNIFAC predicts a larger immiscibility region than observed.
Liquid liquid equilibrium for the ternary system of isopropyl acetate 2 propa...Josemar Pereira da Silva
The document presents experimental data on liquid-liquid equilibrium for the ternary system of isopropyl acetate, 2-propanol, and glycerol at temperatures of 298.15 K, 308.15 K, and 318.15 K under atmospheric pressure. Triangular phase diagrams were obtained at each temperature showing the two-phase region. Distribution coefficients and selectivity parameters were calculated to evaluate glycerol's capacity as an extractive solvent. The NRTL and UNIQUAC models were applied to correlate the experimental data with low deviations.
(Liquid liquid) equilibrium of systems involved in the stepwise ethanolysis o...Josemar Pereira da Silva
The document discusses (liquid + liquid) equilibrium experiments involving systems relevant to the transesterification process for biodiesel production from vegetable oils. Specifically, equilibrium data was measured for systems containing vegetable oils, monoacylglycerols, diacylglycerols, ethyl esters, and ethanol. The experimental results provide information on phase behavior during transesterification and purification steps. The data was also used to evaluate thermodynamic models for predicting multicomponent phase equilibria in these systems.
Liquid liquid equilibrium data for n hexane ethylacetate acetonitrile ternay ...Josemar Pereira da Silva
This document presents experimental liquideliquid equilibrium (LLE) data for the ternary system of n-hexane, ethyl acetate, and acetonitrile measured at 298.15 K, 308.15 K, and 318.15 K. The NRTL and UNIQUAC models were used to correlate the LLE data. Distribution coefficients and separation factors were also calculated from the LLE data, showing that acetonitrile could be used to extract ethyl acetate from n-hexane.
Liquid liquid equilibria data for ethylbenzene or p xylene with alkane and 1 ...Josemar Pereira da Silva
This document presents experimental liquid-liquid equilibrium data for ternary mixtures of 1-butylpyridinium nitrate ionic liquid, various alkanes (heptane, octane, decane), and either ethylbenzene or p-xylene at 298.15 K and 89 kPa. The data includes mole fraction measurements of each phase and calculations of distribution ratios and separation factors. Six different ternary systems were studied and the results suggest p-xylene can be separated more easily from alkanes than ethylbenzene.
Influence of temperature on the liquid liquid equilibria of methanol benzene ...Josemar Pereira da Silva
This document summarizes a study on the liquid-liquid equilibria of the ternary system composed of methanol, benzene, and hexane at temperatures of 278.15 K, 283.15 K, and 293.15 K. Equilibrium data including component mass fractions in each phase are reported for the three temperatures. The data are compared to literature data and correlated using various models including Othmer and Tobias, NRTL, and UNIFAC. The results show that temperature influences the liquid-liquid equilibrium behavior of the system.
Isobaric vapor liquid equilibrium for binary mixtures of 3 methyl 1 butanol 3...Josemar Pereira da Silva
This document presents experimental vapor-liquid equilibrium data for the binary mixtures of 3-methyl-1-butanol + 3-methyl-1-butyl ethanoate and 1-pentanol + pentyl ethanoate at 101.3 kPa. Vapor pressure measurements were made for the pure components using a Swiestoslawski apparatus. Vapor-liquid equilibrium data were obtained using an equilibrium still. Activity coefficients were calculated and correlated using thermodynamic models like Wilson, NRTL, and UNIQUAC. The mixtures were found to not form azeotropes.
Investigation on thermodynamics in separation for ethylene glycol neopentyl g...Josemar Pereira da Silva
The document investigates azeotropic distillation for separating an ethylene glycol (EG) and neopentyl glycol (NPG) system. Vapor-liquid equilibrium data was measured for the EG-NPG and NPG-para-xylene systems under atmospheric pressure. The data was found to be thermodynamically consistent and well-correlated by models like NRTL and UNIQUAC. An azeotropic distillation process was designed using para-xylene as an entrainer that can produce EG and NPG at 99.9% purity with minimal cost.
This document presents a new correlation for predicting the thermal conductivity of liquids based on their molar polarization. The correlation relates two parameters (A and b) from an existing thermal conductivity equation to molar polarization. Molar polarization takes into account molecular structure, polarity, and temperature effects. Experimental thermal conductivity data for various substances was used to develop a correlation between the parameters and molar polarization. The new correlation was found to predict thermal conductivity with average errors less than 5% for most substances tested, outperforming some existing methods.
This document presents a new two-parameter model for predicting the thermal conductivity of liquids. The model is derived based on theories of molecular dynamics, liquid free volume, and the authors' previous model for estimating heat of vaporization. Thermal conductivity data for 68 liquids over a wide range of temperatures are fitted using the model and compared to other existing models, showing good agreement. Key parameters A and B are determined for different liquids to allow calculation of thermal conductivity at any temperature within the studied ranges.
Este documento apresenta um estudo sobre escoamento bifásico líquido-gás em tubulações
horizontais utilizando a correlação de Lockhart & Martinelli e simulações computacionais. Foram
analisados os regimes borbulhado e estratificado para misturas água-ar a 25°C em tubo de 1 polegada.
As simulações computacionais foram realizadas no código ANSYS CFX e comparadas às previsões da
correlação de Lockhart & Martinelli, encontrando-se desvios de até 50% entre os métodos.
Este documento apresenta a tese de doutorado de Cristiano Bigonha Tibiriçá sobre o estudo experimental e teórico da transferência de calor e do fluxo crítico durante a ebulição convectiva em microcanais. O trabalho analisa dados obtidos em bancadas experimentais construídas para este fim e desenvolve modelos considerando os padrões de escoamento observados em microcanais.
On criteria for occurence of azeotropes in isothermal and isobraric binary sy...Josemar Pereira da Silva
The document discusses criteria for predicting the occurrence of azeotropes in binary systems under isothermal and isobaric conditions.
For isothermal systems, the criteria are expressed in terms of the activity coefficients at infinite dilution (γ∞). For a positive azeotrope, γ∞ of the less volatile component must be greater than or equal to the ratio of the saturation vapor pressures. For a negative azeotrope, γ∞ of the more volatile component must be less than or equal to the inverse of the ratio.
For isobaric systems, the criteria involve γ∞ evaluated at the boiling points. For a minimum-boiling azeotrope
This document is a doctoral thesis submitted by Efstathios Skouras-Iliopoulos for the degree of Doctor of Engineering at the Norwegian University of Science and Technology. The thesis focuses on feasibility and operation aspects of heterogeneous azeotropic (heteroazeotropic) distillation in batch distillation columns. Both conventional batch columns (rectifiers) and novel multivessel column configurations are considered for separating azeotropic mixtures. Dynamic simulations are used to analyze time requirements and separation performance of different column configurations for various mixture types. The thesis aims to provide a deeper understanding of heteroazeotropic batch distillation and develop guidelines for assessing feasibility and practical operation of such processes.
This document presents a new two-parameter model for predicting the thermal conductivity of liquids. The model is derived based on theories of molecular dynamics, liquid free volume, and the authors' previous model for estimating heat of vaporization. Thermal conductivity data for 68 liquids over a wide range of temperatures are fitted using the model and compared to other existing models, showing good agreement. Key parameters A and B are determined for different liquids to allow calculation of thermal conductivity based on fundamental physical property data.
Production of-n-propyl-acetate-by-reactive-distillation-experimental-and-theo...Josemar Pereira da Silva
This document summarizes the first steps in developing a catalytic reactive distillation process for producing n-propyl acetate. Kinetic experiments were conducted to determine the reaction rates for homogeneous and heterogeneous catalysis. Pilot plant experiments were also performed using a homogeneous strong acid catalyst in a packed column with a top-column decanter. Simulation results matched experimental data well when accounting for non-ideal thermodynamics. Several process configurations were identified that could dramatically increase alcohol conversion and n-propyl acetate purity by adding a stripping section. The best startup strategy was determined to involve an initial charging of the two-phase top product to achieve steady-state conditions most quickly.
This document summarizes the topological structure of ternary residue curve maps, which describe the dynamics of ternary distillation processes. It introduces differential equations that model ternary distillation and place a meaningful structure on ternary phase diagrams. By recognizing this structure is subject to the Poincaré-Hopf index theorem, the authors obtained a topological relationship between azeotropes and pure components in ternary mixtures. This relationship provides useful information about ternary mixture distillation behavior and predicts situations where ternary azeotropes cannot occur.
The document describes combining the UNIWAALS equation of state with an algorithm for calculating simultaneous chemical and phase equilibria (CPE). This allows CPE calculations for reactive systems containing gases, polar compounds, or both over a broad range of temperatures and pressures using a single thermodynamic model. Examples are given applying the new procedure to different multicomponent reaction systems. The results show good agreement with previous calculations using a different equation of state, demonstrating the capability of performing CPE calculations with UNIWAALS.
UNLOCKING HEALTHCARE 4.0: NAVIGATING CRITICAL SUCCESS FACTORS FOR EFFECTIVE I...amsjournal
The Fourth Industrial Revolution is transforming industries, including healthcare, by integrating digital,
physical, and biological technologies. This study examines the integration of 4.0 technologies into
healthcare, identifying success factors and challenges through interviews with 70 stakeholders from 33
countries. Healthcare is evolving significantly, with varied objectives across nations aiming to improve
population health. The study explores stakeholders' perceptions on critical success factors, identifying
challenges such as insufficiently trained personnel, organizational silos, and structural barriers to data
exchange. Facilitators for integration include cost reduction initiatives and interoperability policies.
Technologies like IoT, Big Data, AI, Machine Learning, and robotics enhance diagnostics, treatment
precision, and real-time monitoring, reducing errors and optimizing resource utilization. Automation
improves employee satisfaction and patient care, while Blockchain and telemedicine drive cost reductions.
Successful integration requires skilled professionals and supportive policies, promising efficient resource
use, lower error rates, and accelerated processes, leading to optimized global healthcare outcomes.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
1. Optimization Toolbox
fmincon
Find a minimum of a constrained nonlinear multivariable function
subject to
where x, b, beq, lb, and ub are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions that return
vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions.
Syntax
x = fmincon(fun,x0,A,b)
x = fmincon(fun,x0,A,b,Aeq,beq)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options,P1,P2, ...)
[x,fval] = fmincon(...)
[x,fval,exitflag] = fmincon(...)
[x,fval,exitflag,output] = fmincon(...)
[x,fval,exitflag,output,lambda] = fmincon(...)
[x,fval,exitflag,output,lambda,grad] = fmincon(...)
[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(...)
Description
fmincon finds a constrained minimum of a scalar function of several variables starting at an initial
estimate. This is generally referred to as constrained nonlinear optimization or nonlinear programming.
x = fmincon(fun,x0,A,b) starts at x0 and finds a minimum x to the function described in fun
subject to the linear inequalities A*x <= b. x0 can be a scalar, vector, or matrix.
x = fmincon(fun,x0,A,b,Aeq,beq) minimizes fun subject to the linear equalities
Aeq*x = beq as well as A*x <= b. Set A=[] and b=[] if no inequalities exist.
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the
design variables, x, so that the solution is always in the range lb <= x <= ub . Set Aeq=[] and
beq=[] if no equalities exist.
2. x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the
nonlinear inequalities c(x) or equalities ceq(x) defined in nonlcon. fmincon optimizes such that
c(x) <= 0 and ceq(x) = 0. Set lb=[] and/or ub=[] if no bounds exist.
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the
optimization parameters specified in the structure options. Use optimset to set these parameters.
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options,P1,P2,...) passes the
problem-dependent parametersP1, P2, etc., directly to the functions fun and nonlcon. Pass empty
matrices as placeholders for A, b, Aeq, beq, lb, ub, nonlcon, and options if these arguments are
not needed.
[x,fval] = fmincon(...) returns the value of the objective functionfun at the solution x.
[x,fval,exitflag] = fmincon(...) returns a valueexitflag that describes the exit
condition of fmincon.
[x,fval,exitflag,output] = fmincon(...) returns a structure output with information
about the optimization.
[x,fval,exitflag,output,lambda] = fmincon(...) returns a structure lambda whose
fields contain the Lagrange multipliers at the solution x.
[x,fval,exitflag,output,lambda,grad] = fmincon(...) returns the value of the gradient
of fun at the solution x.
[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(...) returns the value of
the Hessian of fun at the solution x.
Input Arguments
Function Arguments contains general descriptions of arguments passed in to fmincon. This
"Arguments" section provides function-specific details for fun, nonlcon, and options:
3. fun The function to be minimized. fun is a function that accepts a scalar x and returns a
scalar f, the objective function evaluated atx. The function fun can be specified as a
function handle.
x = fmincon(@myfun,x0,A,b)
where myfun is a MATLAB function such as
function f = myfun(x)
f = ... % Compute function value at x
fun can also be an inline object.
x = fmincon(inline('norm(x)^2'),x0,A,b);
If the gradient offun can also be computed and the GradObj parameter is 'on', as set
by
options = optimset('GradObj','on')
then the function fun must return, in the second output argument, the gradient value g, a
vector, at x. Note that by checking the value of nargout the function can avoid
computing g when fun is called with only one output argument (in the case where the
optimization algorithm only needs the value of f but not g).
function [f,g] = myfun(x)
f = ... % Compute the function value at x
if nargout > 1 % fun called with two output arguments
g = ... % Compute the gradient evaluated at x
end
The gradient consists of the partial derivatives off at the point x. That is, the ith
component of g is the partial derivative off with respect to the ith component of x.
If the Hessian matrix can also be computed and the Hessian parameter is 'on', i.e.,
options = optimset('Hessian','on') , then the function fun must return the
Hessian value H, a symmetric matrix, at x in a third output argument. Note that by
checking the value of nargout we can avoid computing H when fun is called with only
one or two output arguments (in the case where the optimization algorithm only needs the
values off and g but not H).
function [f,g,H] = myfun(x)
f = ... % Compute the objective function value at x
if nargout > 1 % fun called with two output arguments
g = ... % Gradient of the function evaluated at x
if nargout > 2
H = ... % Hessian evaluated at x
end
The Hessian matrix is the second partial derivatives matrix of f at the point x. That is, the
(i,j)th component of H is the second partial derivative off with respect to xi and xj,
. The Hessian is by definition a symmetric matrix.
4. nonlcon The function that computes the nonlinear inequality constraints c(x)<= 0 and the
nonlinear equality constraints ceq(x) = 0. The function nonlcon accepts a vector x
and returns two vectors c and ceq. The vector c contains the nonlinear inequalities
evaluated atx, and ceq contains the nonlinear equalities evaluated at x. The function
nonlcon can be specified as a function handle.
x = fmincon(@myfun,x0,A,b,Aeq,beq,lb,ub,@mycon)
where mycon is a MATLAB function such as
function [c,ceq] = mycon(x)
c = ... % Compute nonlinear inequalities at x.
ceq = ... % Compute nonlinear equalities at x.
If the gradients of the constraints can also be computed and the GradConstr parameter
is 'on', as set by
options = optimset('GradConstr','on')
then the function nonlcon must also return, in the third and fourth output arguments, GC,
the gradient of c(x), and GCeq, the gradient of ceq(x). Note that by checking the value
of nargout the function can avoid computing GC and GCeq when nonlcon is called with
only two output arguments (in the case where the optimization algorithm only needs the
values ofc and ceq but not GC and GCeq).
function [c,ceq,GC,GCeq] = mycon(x)
c = ... % Nonlinear inequalities at x
ceq = ... % Nonlinear equalities at x
if nargout > 2 % nonlcon called with 4 outputs
GC = ... % Gradients of the inequalities
GCeq = ... % Gradients of the equalities
end
If nonlcon returns a vector c of m components and x has length n, where n is the length
of x0, then the gradient GC of c(x) is an n-by-m matrix, where GC(i,j) is the partial
derivative ofc(j) with respect to x(i) (i.e., the jth column of GC is the gradient of the
jth inequality constraint c(j)). Likewise, if ceq has p components, the gradient GCeq of
ceq(x) is an n-by-p matrix, where GCeq(i,j) is the partial derivative ofceq(j) with
respect to x(i) (i.e., the jth column of GCeq is the gradient of the jth equality
constraint ceq(j)).
options Options provides the function-specific details for the options parameters.
Output Arguments
Function Arguments contains general descriptions of arguments returned by fmincon. This section
provides function-specific details forexitflag, lambda, and output:
5. exitflag Describes the exit condition:
> 0 The function converged to a solutionx.
0 The maximum number of function evaluations or iterations was
exceeded.
< 0 The function did not converge to a solution.
lambda Structure containing the Lagrange multipliers at the solution x (separated by constraint
type). The fields of the structure are:
lower Lower bounds lb
upper Upper bounds ub
ineqlin Linear inequalities
eqlin Linear equalities
ineqnonlin Nonlinear inequalities
eqnonlin Nonlinear equalities
output Structure containing information about the optimization. The fields of the structure are:
iterations Number of iterations taken.
funcCount Number of function evaluations.
algorithm Algorithm used.
cgiterations Number of PCG iterations (large-scale algorithm only).
stepsize Final step size taken (medium-scale algorithm only).
firstorderopt Measure of first-order optimality (large-scale algorithm only).
For large-scale bound constrained problems, the first-order
optimality is the infinity norm of v.*g, where v is defined as in Box
Constraints, and g is the gradient.
For large-scale problems with only linear equalities, the first-order
optimality is the infinity norm of the projected gradient (i.e. the
gradient projected onto the nullspace of Aeq).
Options
Optimization options parameters used by fmincon. Some parameters apply to all algorithms, some are
only relevant when using the large-scale algorithm, and others are only relevant when using the
medium-scale algorithm.You can use optimset to set or change the values of these fields in the
parameters structure, options. See Optimization Parameters, for detailed information.
6. We start by describing the LargeScale option since it states a preference for which algorithm to use.
It is only a preference since certain conditions must be met to use the large-scale algorithm. For
fmincon, you must provide the gradient (see the description of fun above to see how) or else the
medium-scale algorithm is used:
LargeScale Use large-scale algorithm if possible when set to 'on'. Use medium-scale algorithm
when set to 'off'.
Medium-Scale and Large-Scale Algorithms. These parameters are used by both the medium-scale
and large-scale algorithms:
Diagnostics Print diagnostic information about the function to be minimized.
Display Level of display. 'off' displays no output; 'iter' displays output at each
iteration; 'final' (default) displays just the final output.
GradObj Gradient for the objective function defined by user. See the description offun above
to see how to define the gradient in fun. You must provide the gradient to use the
large-scale method. It is optional for the medium-scale method.
MaxFunEvals Maximum number of function evaluations allowed.
MaxIter Maximum number of iterations allowed.
TolFun Termination tolerance on the function value.
TolCon Termination tolerance on the constraint violation.
TolX Termination tolerance onx.
Large-Scale Algorithm Only. These parameters are used only by the large-scale algorithm:
Hessian If 'on', fmincon uses a user-defined Hessian (defined infun), or Hessian
information (when usingHessMult), for the objective function. If'off',
fmincon approximates the Hessian using finite differences.
HessMult Function handle for Hessian multiply function. For large-scale structured
problems, this function computes the Hessian matrix product H*Y without
actually forming H. The function is of the form
W = hmfun(Hinfo,Y,p1,p2,...)
7. where Hinfo and the additional parameters p1,p2,... contain the
matrices used to compute H*Y. The first argument must be the same as the
third argument returned by the objective functionfun.
[f,g,Hinfo] = fun(x,p1,p2,...)
The parameters p1,p2,... are the same additional parameters that are
passed to fmincon (and to fun).
fmincon(fun,...,options,p1,p2,...)
Y is a matrix that has the same number of rows as there are dimensions in
the problem. W = H*Y although H is not formed explicitly. fmincon uses
Hinfo to compute the preconditioner.
Note 'Hessian' must be set to 'on' for Hinfo to be passed
from fun to hmfun.
See Nonlinear Minimization with a Dense but Structured Hessian and
Equality Constraints for an example.
HessPattern Sparsity pattern of the Hessian for finite-differencing. If it is not convenient to
compute the sparse Hessian matrix H in fun, the large-scale method in
fmincon can approximate H via sparse finite-differences (of the gradient)
provided thesparsity structure of H -- i.e., locations of the nonzeros -- is
supplied as the value forHessPattern. In the worst case, if the structure is
unknown, you can set HessPattern to be a dense matrix and a full
finite-difference approximation is computed at each iteration (this is the
default). This can be very expensive for large problems so it is usually worth
the effort to determine the sparsity structure.
MaxPCGIter Maximum number of PCG (preconditioned conjugate gradient) iterations (see
the Algorithm section below).
PrecondBandWidth Upper bandwidth of preconditioner for PCG. By default, diagonal
preconditioning is used (upper bandwidth of 0). For some problems,
increasing the bandwidth reduces the number of PCG iterations.
TolPCG Termination tolerance on the PCG iteration.
TypicalX Typical x values.
Medium-Scale Algorithm Only. These parameters are used only by the medium-scale algorithm:
DerivativeCheck Compare user-supplied derivatives (gradients of the objective and constraints)
to finite-differencing derivatives.
DiffMaxChange Maximum change in variables for finite-difference gradients.
DiffMinChange Minimum change in variables for finite-difference gradients.
8. Examples
Find values ofx that minimize , starting at the point x = [10; 10; 10] and subject
to the constraints
First, write an M-file that returns a scalar value f of the function evaluated atx.
function f = myfun(x)
f = -x(1) * x(2) * x(3);
Then rewrite the constraints as both less than or equal to a constant,
Since both constraints are linear, formulate them as the matrix inequality where
Next, supply a starting point and invoke an optimization routine.
x0 = [10; 10; 10]; % Starting guess at the solution
[x,fval] = fmincon(@myfun,x0,A,b)
After 66 function evaluations, the solution is
x =
24.0000
12.0000
12.0000
where the function value is
fval =
-3.4560e+03
and linear inequality constraints evaluate to be <= 0
A*x-b=
-72
0
Notes
9. Large-Scale Optimization. To use the large-scale method, the gradient must be provided infun (and
the GradObj parameter is set to 'on'). A warning is given if no gradient is provided and the
LargeScale parameter is not 'off'. The function fmincon permits g(x) to be an approximate
gradient but this option is not recommended; the numerical behavior of most optimization codes is
considerably more robust when the true gradient is used.
The large-scale method in fmincon is most effective when the matrix of second derivatives, i.e., the
Hessian matrix H(x), is also computed. However, evaluation of the true Hessian matrix is not required.
For example, if you can supply the Hessian sparsity structure (using the HessPattern parameter in
options), then fmincon computes a sparse finite-difference approximation to H(x).
If x0 is not strictly feasible, fmincon chooses a new strictly feasible (centered) starting point.
If components of x have no upper (or lower) bounds, thenfmincon prefers that the corresponding
components of ub (or lb) be set to Inf (or -Inf for lb) as opposed to an arbitrary but very large
positive (or negative in the case of lower bounds) number.
Several aspects of linearly constrained minimization should be noted:
A dense (or fairly dense) column of matrix Aeq can result in considerable fill and computational
cost.
fmincon removes (numerically) linearly dependent rows inAeq; however, this process involves
repeated matrix factorizations and therefore can be costly if there are many dependencies.
Each iteration involves a sparse least-squares solve with matrix
where RT is the Cholesky factor of the preconditioner. Therefore, there is a potential conflict
between choosing an effective preconditioner and minimizing fill in .
Medium-Scale Optimization. Better numerical results are likely if you specify equalities explicitly
using Aeq and beq, instead of implicitly using lb and ub.
If equality constraints are present and dependent equalities are detected and removed in the quadratic
subproblem, 'dependent' is printed under the Procedures heading (when you ask for output by
setting the Display parameter to'iter'). The dependent equalities are only removed when the
equalities are consistent. If the system of equalities is not consistent, the subproblem is infeasible and
'infeasible' is printed under the Procedures heading.
Algorithm
Large-Scale Optimization. By default fmincon will choose the large-scale algorithm if the user
supplies the gradient in fun (and GradObj is 'on' in options) and if only upper and lower bounds
exist or only linear equality constraints exist. This algorithm is a subspace trust region method and is
based on the interior-reflective Newton method described in[1], [2]. Each iteration involves the
approximate solution of a large linear system using the method of preconditioned conjugate gradients
(PCG). See the trust-region and preconditioned conjugate gradient method descriptions in the
Large-Scale Algorithms chapter.
10. Medium-Scale Optimization. fmincon uses a Sequential Quadratic Programming (SQP) method. In
this method, a Quadratic Programming (QP) subproblem is solved at each iteration. An estimate of the
Hessian of the Lagrangian is updated at each iteration using the BFGS formula (see fminunc,
references [7], [8]).
A line search is performed using a merit function similar to that proposed by [4], [5], and [6]. The QP
subproblem is solved using an active set strategy similar to that described in [3]. A full description of this
algorithm is found in Constrained Optimization in "Introduction to Algorithms."
See also SQP Implementation in "Introduction to Algorithms" for more details on the algorithm used.
Diagnostics
Large-Scale Optimization. The large-scale code does not allow equal upper and lower bounds. For
example if lb(2)==ub(2) , then fmincon gives the error
Equal upper and lower bounds not permitted in this large-scale
method.
Use equality constraints and the medium-scale method instead.
If you only have equality constraints you can still use the large-scale method. But if you have both
equalities and bounds, you must use the medium-scale method.
Limitations
The function to be minimized and the constraints must both be continuous. fmincon may only give
local solutions.
When the problem is infeasible, fmincon attempts to minimize the maximum constraint value.
The objective function and constraint function must be real-valued, that is they cannot return complex
values.
Large-Scale Optimization. To use the large-scale algorithm, the user must supply the gradient in
fun (and GradObj must be set 'on' in options) , and only upper and lower bounds constraints may
be specified, or only linear equality constraints must exist and Aeq cannot have more rows than
columns. Aeq is typically sparse. See Table 2-4, Large-Scale Problem Coverage and Requirements,, for
more information on what problem formulations are covered and what information must be provided.
Currently, if the analytical gradient is provided in fun, the options parameter DerivativeCheck
cannot be used with the large-scale method to compare the analytic gradient to the finite-difference
gradient. Instead, use the medium-scale method to check the derivative with options parameter
MaxIter set to 0 iterations. Then run the problem with the large-scale method.
See Also
@ (function_handle ), fminbnd, fminsearch, fminunc, optimset
References
[1] Coleman, T.F. and Y. Li, "An Interior, Trust Region Approach for Nonlinear Minimization Subject to
Bounds," SIAM Journal on Optimization, Vol. 6, pp. 418-445, 1996.
11. [2] Coleman, T.F. and Y. Li, "On the Convergence of Reflective Newton Methods for Large-Scale
Nonlinear Minimization Subject to Bounds," Mathematical Programming, Vol. 67, Number 2, pp.
189-224, 1994.
[3] Gill, P.E., W. Murray, and M.H. Wright, Practical Optimization, Academic Press, London, 1981.
[4] Han, S.P., "A Globally Convergent Method for Nonlinear Programming," Journal of Optimization
Theory and Applications, Vol. 22, p. 297, 1977.
[5] Powell, M.J.D., "A Fast Algorithm for Nonlineary Constrained Optimization Calculations," Numerical
Analysis, ed. G.A. Watson, Lecture Notes in Mathematics, Springer Verlag, Vol. 630, 1978.
[6] Powell, M.J.D., "The Convergence of Variable Metric Methods For Nonlinearly Constrained
Optimization Calculations," Nonlinear Programming 3, (O.L. Mangasarian, R.R. Meyer, and S.M.
Robinson, eds.) Academic Press, 1978.
fminbnd fminimax