Lockhart and Johnson (1996) define optimization as “the process of finding the most effective or favorable value or condition” (p. 610). The purpose of optimization is to achieve the “best” design relative to a set of prioritized criteria or constraints.Within the traditional engineering disciplines, optimization techniques are commonly employed for a variety of problems, including: Product-Mix Problems. Determine the mix of products in a factory that will make the best use of machines, labor resources, raw materials, while maximizing the companies profitsOptimization involves the selection of the “best” solution from among the set of candidate solutions. The degree of goodness of the solution is quantified using an objective function (e.g., cost) which is to be minimized or maximized.Optimization problem: Maximizing or minimizing some function relative to some set,
often representing a range of choices available in a certain situation. The function
allows comparison of the different choices for determining which might be “best.”
Common applications: Minimal cost, maximal profit, minimal error, optimal design,
optimal management, variational principles.
Goals of the subject: The understanding of
Modeling issues—
What to look for in setting up an optimization problem?
What features are advantageous or disadvantageous?
What devices/tricks of formulation are available?
How can problems usefully be categorized?
Analysis of solutions—
What is meant by a “solution?”
When do solutions exist, and when are they unique?
How can solutions be recognized and characterized?
What happens to solutions under perturbations?
Numerical methods—
How can solutions be determined by iterative schemes of computation?
What modes of local simplification of a problem are convenient/appropriate?
How can different solution techniques be compared and evaluated?Distinguishing features of optimization as a mathematical discipline:
descriptive −→ prescriptive
equations −→ inequalities
linear/nonlinear −→ convex/nonconvex
differential calculus −→ subdifferential calculus
1
Finite-dimensional optimization: The case where a choice corresponds to selecting
the values of a finite number of real variables, called decision variables. For general
purposes the decision variables may be denoted by x1, . . . , xn and each possible choice
therefore identified with a point x = (x1, . . . , xn) in the space IRn
. This is what we’ll
be focusing on in this course.
Feasible set: The subset C of IRn
representing the allowable choices x = (x1, . . . , xn).
Objective function: The function f0(x) = f0(x1, . . . , xn) that is to be maximized or
minimized over C.
Constraints: Side conditions that are used to specify the feasible set C within IRn
.
Equality constraints: Conditions of the form fi(x) = ci
for certain functions fi on IRn
and constants ci
in IRn
.
Inequality constraints: Conditions of the form fi(x) ≤ ci or fi(x) ≥ ci
for certain
functions fi on IRn
and constants ci
in IR.
Range constarintt
https://telecombcn-dl.github.io/2017-dlai/
Deep learning technologies are at the core of the current revolution in artificial intelligence for multimedia data analysis. The convergence of large-scale annotated datasets and affordable GPU hardware has allowed the training of neural networks for data analysis tasks which were previously addressed with hand-crafted features. Architectures such as convolutional neural networks, recurrent neural networks or Q-nets for reinforcement learning have shaped a brand new scenario in signal processing. This course will cover the basic principles of deep learning from both an algorithmic and computational perspectives.
https://telecombcn-dl.github.io/2018-dlai/
Deep learning technologies are at the core of the current revolution in artificial intelligence for multimedia data analysis. The convergence of large-scale annotated datasets and affordable GPU hardware has allowed the training of neural networks for data analysis tasks which were previously addressed with hand-crafted features. Architectures such as convolutional neural networks, recurrent neural networks or Q-nets for reinforcement learning have shaped a brand new scenario in signal processing. This course will cover the basic principles of deep learning from both an algorithmic and computational perspectives.
https://telecombcn-dl.github.io/2017-dlai/
Deep learning technologies are at the core of the current revolution in artificial intelligence for multimedia data analysis. The convergence of large-scale annotated datasets and affordable GPU hardware has allowed the training of neural networks for data analysis tasks which were previously addressed with hand-crafted features. Architectures such as convolutional neural networks, recurrent neural networks or Q-nets for reinforcement learning have shaped a brand new scenario in signal processing. This course will cover the basic principles of deep learning from both an algorithmic and computational perspectives.
https://telecombcn-dl.github.io/2018-dlai/
Deep learning technologies are at the core of the current revolution in artificial intelligence for multimedia data analysis. The convergence of large-scale annotated datasets and affordable GPU hardware has allowed the training of neural networks for data analysis tasks which were previously addressed with hand-crafted features. Architectures such as convolutional neural networks, recurrent neural networks or Q-nets for reinforcement learning have shaped a brand new scenario in signal processing. This course will cover the basic principles of deep learning from both an algorithmic and computational perspectives.
Overview on Optimization algorithms in Deep LearningKhang Pham
Overview on function optimization in general and in deep learning. The slides cover from basic algorithms like batch gradient descent, stochastic gradient descent to the state of art algorithm like Momentum, Adagrad, RMSprop, Adam.
Nearest neighbor models are conceptually just about the simplest kind of model possible. The problem is that they generally aren’t feasible to apply. Or at least, they weren’t feasible until the advent of Big Data techniques. These slides will describe some of the techniques used in the knn project to reduce thousand-year computations to a few hours. The knn project uses the Mahout math library and Hadoop to speed up these enormous computations to the point that they can be usefully applied to real problems. These same techniques can also be used to do real-time model scoring.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
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Overview on Optimization algorithms in Deep LearningKhang Pham
Overview on function optimization in general and in deep learning. The slides cover from basic algorithms like batch gradient descent, stochastic gradient descent to the state of art algorithm like Momentum, Adagrad, RMSprop, Adam.
Nearest neighbor models are conceptually just about the simplest kind of model possible. The problem is that they generally aren’t feasible to apply. Or at least, they weren’t feasible until the advent of Big Data techniques. These slides will describe some of the techniques used in the knn project to reduce thousand-year computations to a few hours. The knn project uses the Mahout math library and Hadoop to speed up these enormous computations to the point that they can be usefully applied to real problems. These same techniques can also be used to do real-time model scoring.
Similar to cos323_s06_lecture03_optimization.ppt (20)
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
2. Ingredients
• Objective function
• Variables
• Constraints
Find values of the variables
that minimize or maximize the objective function
while satisfying the constraints
3. Different Kinds of Optimization
Figure from: Optimization Technology Center
http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/
4. Different Optimization Techniques
• Algorithms have very different flavor depending
on specific problem
– Closed form vs. numerical vs. discrete
– Local vs. global minima
– Running times ranging from O(1) to NP-hard
• Today:
– Focus on continuous numerical methods
5. Optimization in 1-D
• Look for analogies to bracketing in root-finding
• What does it mean to bracket a minimum?
(xleft, f(xleft))
(xright, f(xright))
(xmid, f(xmid))
xleft < xmid < xright
f(xmid) < f(xleft)
f(xmid) < f(xright)
6. Optimization in 1-D
• Once we have these properties, there is at least
one local minimum between xleft and xright
• Establishing bracket initially:
– Given xinitial, increment
– Evaluate f(xinitial), f(xinitial+increment)
– If decreasing, step until find an increase
– Else, step in opposite direction until find an increase
– Grow increment at each step
• For maximization: substitute –f for f
7. Optimization in 1-D
• Strategy: evaluate function at some xnew
(xleft, f(xleft))
(xright, f(xright))
(xmid, f(xmid))
(xnew, f(xnew))
8. Optimization in 1-D
• Strategy: evaluate function at some xnew
– Here, new “bracket” points are xnew, xmid, xright
(xleft, f(xleft))
(xright, f(xright))
(xmid, f(xmid))
(xnew, f(xnew))
9. Optimization in 1-D
• Strategy: evaluate function at some xnew
– Here, new “bracket” points are xleft, xnew, xmid
(xleft, f(xleft))
(xright, f(xright))
(xmid, f(xmid))
(xnew, f(xnew))
10. Optimization in 1-D
• Unlike with root-finding, can’t always guarantee
that interval will be reduced by a factor of 2
• Let’s find the optimal place for xmid, relative to
left and right, that will guarantee same factor of
reduction regardless of outcome
12. Golden Section Search
• To assure same interval, want = 1–2
• So,
• This is the “golden ratio” = 0.618…
• So, interval decreases by 30% per iteration
– Linear convergence
2
1
5
13. Error Tolerance
• Around minimum, derivative = 0, so
• Rule of thumb: pointless to ask for more
accuracy than sqrt()
– Can use double precision if you want a single-
precision result (and/or have single-precision data)
~
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14. Faster 1-D Optimization
• Trade off super-linear convergence for
worse robustness
– Combine with Golden Section search for safety
• Usual bag of tricks:
– Fit parabola through 3 points, find minimum
– Compute derivatives as well as positions, fit cubic
– Use second derivatives: Newton
19. Newton’s Method
• At each step:
• Requires 1st and 2nd derivatives
• Quadratic convergence
)
(
)
(
1
k
k
k
k
x
f
x
f
x
x
20. Multi-Dimensional Optimization
• Important in many areas
– Fitting a model to measured data
– Finding best design in some parameter space
• Hard in general
– Weird shapes: multiple extrema, saddles,
curved or elongated valleys, etc.
– Can’t bracket
• In general, easier than rootfinding
– Can always walk “downhill”
21. Newton’s Method in
Multiple Dimensions
• Replace 1st derivative with gradient,
2nd derivative with Hessian
2
2
2
2
2
2
)
,
(
y
f
y
x
f
y
x
f
x
f
y
f
x
f
H
f
y
x
f
22. Newton’s Method in
Multiple Dimensions
• Replace 1st derivative with gradient,
2nd derivative with Hessian
• So,
• Tends to be extremely fragile unless function
very smooth and starting close to minimum
)
(
)
(
1
1 k
k
k
k x
f
x
H
x
x
23. Important classification of methods
• Use function + gradient + Hessian (Newton)
• Use function + gradient (most descent methods)
• Use function values only (Nelder-Mead, called also
“simplex”, or “amoeba” method)
24. Steepest Descent Methods
• What if you can’t / don’t want to
use 2nd derivative?
• “Quasi-Newton” methods estimate Hessian
• Alternative: walk along (negative of) gradient…
– Perform 1-D minimization along line passing through
current point in the direction of the gradient
– Once done, re-compute gradient, iterate
27. Conjugate Gradient Methods
• Idea: avoid “undoing”
minimization that’s
already been done
• Walk along direction
• Polak and Ribiere
formula:
k
k
k
k d
g
d
1
1
k
k
k
k
k
g
g
g
g
gk
T
1
T
)
(
1
28. Conjugate Gradient Methods
• Conjugate gradient implicitly obtains
information about Hessian
• For quadratic function in n dimensions, gets
exact solution in n steps (ignoring roundoff error)
• Works well in practice…
29. Value-Only Methods in Multi-Dimensions
• If can’t evaluate gradients, life is hard
• Can use approximate (numerically evaluated)
gradients:
)
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3
2
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2
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f
e
f
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f
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f
x
f
30. Generic Optimization Strategies
• Uniform sampling:
– Cost rises exponentially with # of dimensions
• Simulated annealing:
– Search in random directions
– Start with large steps, gradually decrease
– “Annealing schedule” – how fast to cool?
31. Downhill Simplex Method (Nelder-Mead)
• Keep track of n+1 points in n dimensions
– Vertices of a simplex (triangle in 2D
tetrahedron in 3D, etc.)
• At each iteration: simplex can move,
expand, or contract
– Sometimes known as amoeba method:
simplex “oozes” along the function
32. Downhill Simplex Method (Nelder-Mead)
• Basic operation: reflection
worst point
(highest function value)
location probed by
reflection step
33. Downhill Simplex Method (Nelder-Mead)
• If reflection resulted in best (lowest) value so far,
try an expansion
• Else, if reflection helped at all, keep it
location probed by
expansion step
34. Downhill Simplex Method (Nelder-Mead)
• If reflection didn’t help (reflected point still worst)
try a contraction
location probed by
contration step
35. Downhill Simplex Method (Nelder-Mead)
• If all else fails shrink the simplex around
the best point
36. Downhill Simplex Method (Nelder-Mead)
• Method fairly efficient at each iteration
(typically 1-2 function evaluations)
• Can take lots of iterations
• Somewhat flakey – sometimes needs restart
after simplex collapses on itself, etc.
• Benefits: simple to implement, doesn’t need
derivative, doesn’t care about function
smoothness, etc.
37. Rosenbrock’s Function
• Designed specifically for testing
optimization techniques
• Curved, narrow valley
2
2
2
)
1
(
)
(
100
)
,
( x
x
y
y
x
f
38. Constrained Optimization
• Equality constraints: optimize f(x)
subject to gi(x)=0
• Method of Lagrange multipliers: convert to a
higher-dimensional problem
• Minimize w.r.t.
)
(
)
( x
g
x
f i
i
)
;
( 1
1 k
n
x
x
39. Constrained Optimization
• Inequality constraints are harder…
• If objective function and constraints all linear,
this is “linear programming”
• Observation: minimum must lie at corner of
region formed by constraints
• Simplex method: move from vertex to vertex,
minimizing objective function
41. Global Optimization
• In general, can’t guarantee that you’ve found
global (rather than local) minimum
• Some heuristics:
– Multi-start: try local optimization from
several starting positions
– Very slow simulated annealing
– Use analytical methods (or graphing) to determine
behavior, guide methods to correct neighborhoods