Probabilistic Control of Uncertain Linear Systems Using Stochastic ReachabilityLeo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of discrete-time uncertain systems based on reachable set computations in
a stochastic setting. For given Gaussian distributions of the initial states and disturbances, state
sets wich are reachable to a chosen confidence level under the effect of time-variant control laws
are computed by using principles of the ellipsoidal calculus. The proposed algorithm iterates over
LMI-constrained semi-definite programming problems to compute probabilistically stabilizing
controllers, while ellipsoidal input constraints are considered. An example for illustration is included.
Introduction of Quantum Annealing and D-Wave MachinesArithmer Inc.
This slide was used for Arithmer seminar in April 2021, by Dr. Yuki Bando.
It is for introduction of quantum computer, D-wave series, and its application to optimization problems in industry.
"Arithmer Seminar" is weekly held, where professionals from within and outside our company give lectures on their respective expertise.
The slides are made by the lecturer from outside our company, and shared here with his/her permission.
Arithmer株式会社は東京大学大学院数理科学研究科発の数学の会社です。私達は現代数学を応用して、様々な分野のソリューションに、新しい高度AIシステムを導入しています。AIをいかに上手に使って仕事を効率化するか、そして人々の役に立つ結果を生み出すのか、それを考えるのが私たちの仕事です。
Arithmer began at the University of Tokyo Graduate School of Mathematical Sciences. Today, our research of modern mathematics and AI systems has the capability of providing solutions when dealing with tough complex issues. At Arithmer we believe it is our job to realize the functions of AI through improving work efficiency and producing more useful results for society.
Probabilistic Control of Uncertain Linear Systems Using Stochastic ReachabilityLeo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of discrete-time uncertain systems based on reachable set computations in
a stochastic setting. For given Gaussian distributions of the initial states and disturbances, state
sets wich are reachable to a chosen confidence level under the effect of time-variant control laws
are computed by using principles of the ellipsoidal calculus. The proposed algorithm iterates over
LMI-constrained semi-definite programming problems to compute probabilistically stabilizing
controllers, while ellipsoidal input constraints are considered. An example for illustration is included.
Introduction of Quantum Annealing and D-Wave MachinesArithmer Inc.
This slide was used for Arithmer seminar in April 2021, by Dr. Yuki Bando.
It is for introduction of quantum computer, D-wave series, and its application to optimization problems in industry.
"Arithmer Seminar" is weekly held, where professionals from within and outside our company give lectures on their respective expertise.
The slides are made by the lecturer from outside our company, and shared here with his/her permission.
Arithmer株式会社は東京大学大学院数理科学研究科発の数学の会社です。私達は現代数学を応用して、様々な分野のソリューションに、新しい高度AIシステムを導入しています。AIをいかに上手に使って仕事を効率化するか、そして人々の役に立つ結果を生み出すのか、それを考えるのが私たちの仕事です。
Arithmer began at the University of Tokyo Graduate School of Mathematical Sciences. Today, our research of modern mathematics and AI systems has the capability of providing solutions when dealing with tough complex issues. At Arithmer we believe it is our job to realize the functions of AI through improving work efficiency and producing more useful results for society.
Probabilistic Control of Switched Linear Systems with Chance ConstraintsLeo Asselborn
An approach to algorithmically synthesize control
strategies for set-to-set transitions of uncertain discrete-time
switched linear systems based on a combination of tree search
and reachable set computations in a stochastic setting is
proposed in this presentation. The initial state and disturbances
are assumed to be Gaussian distributed, and a time-variant
hybrid control law stabilizes the system towards a goal set.
The algorithmic solution computes sequences of discrete states
via tree search and the continuous controls are obtained
from solving embedded semi-definite programs (SDP). These
program taking polytopic input constraints as well as timevarying
probabilistic state constraints into account. An example
for demonstrating the principles of the solution procedure with
focus on handling the chance constraints is included.
Given two integer arrays val[0...n-1] and wt[0...n-1] that represents values and weights associated with n items respectively. Find out the maximum value subset of val[] such that sum of the weights of this subset is smaller than or equal to knapsack capacity W. Here the BRANCH AND BOUND ALGORITHM is discussed .
Control of Discrete-Time Piecewise Affine Probabilistic Systems using Reachab...Leo Asselborn
This presentation proposes an algorithmic approach to
synthesize stabilizing control laws for discrete-time piecewise
affine probabilistic (PWAP) systems based on computations of
probabilistic reachable sets. The considered class of systems
contains probabilistic components (with Gaussian distribution)
modeling additive disturbances and state initialization. The
probabilistic reachable state sets contain all states that are
reachable with a given confidence level under the effect of
time-variant control laws. The control synthesis uses principles
of the ellipsoidal calculus, and it considers that the system
parametrization depends on the partition of the state space. The
proposed algorithm uses LMI-constrained semi-definite programming
(SDP) problems to compute stabilizing controllers,
while polytopic input constraints and transitions between regions
of the state space are considered. The formulation of
the SDP is adopted from a previous work in [1] for switched
systems, in which the switching of the continuous dynamics
is triggered by a discrete input variable. Here, as opposed
to [1], the switching occurs autonomously and an algorithmic
procedure is suggested to synthesis a stabilizing controller. An
example for illustration is included.
Cryptanalysis with a Quantum Computer - An Exposition on Shor's Factoring Alg...Daniel Hutama
Integer factorization is a problem that has been studied by mathematicians for centuries, but has yet to see an efficient classical solution. The apparent intractability of the factorization problem has become the cornerstone of several cryptosystems, such as the widely used RSA encryption scheme for securing financial transactions and communications.
In this presentation, we show an in-depth study of quantum circuit designs for a quantum computer running Shor's algorithm. In particular, we present a classical-based reversible quantum circuit design of Vedral et. al., and a Fourier space circuit designed by Draper and Beauregard. Included in the appendix are detailed descriptions of Shor's full algorithm and a fully worked (classically simulated) example for factoring a 5-bit semiprime number.
Readers should have a basic knowledge of quantum computing concepts, such as qubits, quantum logic gates, entanglement, and their mathematical descriptions.
Gradient descent optimization with simple examples. covers sgd, mini-batch, momentum, adagrad, rmsprop and adam.
Made for people with little knowledge of neural network.
Bayesian Estimation For Modulated Claim HedgingIJERA Editor
The purpose of this paper is to establish a general super hedging formula under a pricing set Q. We will compute
the price and the strategies for hedging an European claim and simulate that using different approaches including
Dirichlet priors. We study Dirichlet processes centered around the distribution of continuous-time stochastic
processes such as a continuous time Markov chain. We assume that the prior distribution of the unobserved
Markov chain driving by the drift and volatility parameters of the geometric Brownian motion (GBM) is a
Dirichlet process. We propose an estimation method based on Gibbs sampling.
Shor's algorithm is for quantum computer. Using this algorithm any arbitrarily large number can be factored in polynomial time. which is not possible in classical computer
Probabilistic Control of Switched Linear Systems with Chance ConstraintsLeo Asselborn
An approach to algorithmically synthesize control
strategies for set-to-set transitions of uncertain discrete-time
switched linear systems based on a combination of tree search
and reachable set computations in a stochastic setting is
proposed in this presentation. The initial state and disturbances
are assumed to be Gaussian distributed, and a time-variant
hybrid control law stabilizes the system towards a goal set.
The algorithmic solution computes sequences of discrete states
via tree search and the continuous controls are obtained
from solving embedded semi-definite programs (SDP). These
program taking polytopic input constraints as well as timevarying
probabilistic state constraints into account. An example
for demonstrating the principles of the solution procedure with
focus on handling the chance constraints is included.
Given two integer arrays val[0...n-1] and wt[0...n-1] that represents values and weights associated with n items respectively. Find out the maximum value subset of val[] such that sum of the weights of this subset is smaller than or equal to knapsack capacity W. Here the BRANCH AND BOUND ALGORITHM is discussed .
Control of Discrete-Time Piecewise Affine Probabilistic Systems using Reachab...Leo Asselborn
This presentation proposes an algorithmic approach to
synthesize stabilizing control laws for discrete-time piecewise
affine probabilistic (PWAP) systems based on computations of
probabilistic reachable sets. The considered class of systems
contains probabilistic components (with Gaussian distribution)
modeling additive disturbances and state initialization. The
probabilistic reachable state sets contain all states that are
reachable with a given confidence level under the effect of
time-variant control laws. The control synthesis uses principles
of the ellipsoidal calculus, and it considers that the system
parametrization depends on the partition of the state space. The
proposed algorithm uses LMI-constrained semi-definite programming
(SDP) problems to compute stabilizing controllers,
while polytopic input constraints and transitions between regions
of the state space are considered. The formulation of
the SDP is adopted from a previous work in [1] for switched
systems, in which the switching of the continuous dynamics
is triggered by a discrete input variable. Here, as opposed
to [1], the switching occurs autonomously and an algorithmic
procedure is suggested to synthesis a stabilizing controller. An
example for illustration is included.
Cryptanalysis with a Quantum Computer - An Exposition on Shor's Factoring Alg...Daniel Hutama
Integer factorization is a problem that has been studied by mathematicians for centuries, but has yet to see an efficient classical solution. The apparent intractability of the factorization problem has become the cornerstone of several cryptosystems, such as the widely used RSA encryption scheme for securing financial transactions and communications.
In this presentation, we show an in-depth study of quantum circuit designs for a quantum computer running Shor's algorithm. In particular, we present a classical-based reversible quantum circuit design of Vedral et. al., and a Fourier space circuit designed by Draper and Beauregard. Included in the appendix are detailed descriptions of Shor's full algorithm and a fully worked (classically simulated) example for factoring a 5-bit semiprime number.
Readers should have a basic knowledge of quantum computing concepts, such as qubits, quantum logic gates, entanglement, and their mathematical descriptions.
Gradient descent optimization with simple examples. covers sgd, mini-batch, momentum, adagrad, rmsprop and adam.
Made for people with little knowledge of neural network.
Bayesian Estimation For Modulated Claim HedgingIJERA Editor
The purpose of this paper is to establish a general super hedging formula under a pricing set Q. We will compute
the price and the strategies for hedging an European claim and simulate that using different approaches including
Dirichlet priors. We study Dirichlet processes centered around the distribution of continuous-time stochastic
processes such as a continuous time Markov chain. We assume that the prior distribution of the unobserved
Markov chain driving by the drift and volatility parameters of the geometric Brownian motion (GBM) is a
Dirichlet process. We propose an estimation method based on Gibbs sampling.
Shor's algorithm is for quantum computer. Using this algorithm any arbitrarily large number can be factored in polynomial time. which is not possible in classical computer
270-1/02-divide-and-conquer_handout.pdf
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Week 2
Divide and Conquer
1 Growth of Functions
2 Divide-and-Conquer
Min-Max-Problem
3 Tutorial
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
General remarks
First we consider an important tool for the analysis of
algorithms: Big-Oh.
Then we introduce an important algorithmic paradigm:
Divide-and-Conquer.
We conclude by presenting and analysing a simple example.
Reading from CLRS for week 2
Chapter 2, Section 3
Chapter 3
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Growth of Functions
A way to describe behaviour of functions in the limit. We
are studying asymptotic efficiency.
Describe growth of functions.
Focus on what’s important by abstracting away low-order
terms and constant factors.
How we indicate running times of algorithms.
A way to compare “sizes” of functions:
O corresponds to ≤
Ω corresponds to ≥
Θ corresponds to =
We consider only functions f , g : N → R≥0.
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
O-Notation
O
(
g(n)
)
is the set of all functions f (n) for which there are
positive constants c and n0 such that
f (n) ≤ cg(n) for all n ≥ n0.
cg(n)
f (n)
n
n0
g(n) is an asymptotic upper bound for f (n).
If f (n) ∈ O(g(n)), we write f (n) = O(g(n)) (we will precisely
explain this soon)
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
O-Notation Examples
2n2 = O(n3), with c = 1 and n0 = 2.
Example of functions in O(n2):
n2
n2 + n
n2 + 1000n
1000n2 + 1000n
Also
n
n/1000
n1.999999
n2/ lg lg lg n
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Ω-Notation
Ω
(
g(n)
)
is the set of all functions f (n) for which there are
positive constants c and n0 such that
f (n) ≥ cg(n) for all n ≥ n0.
cg(n)
f (n)
n
n0
g(n) is an asymptotic lower bound for f (n).
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Ω-Notation Examples
√
n = Ω(lg n), with c = 1 and n0 = 16.
Example of functions in Ω(n2):
n2
n2 + n
n2 − n
1000n2 + 1000n
1000n2 − 1000n
Also
n3
n2.0000001
n2 lg lg lg n
22
n
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Θ-Notation
Θ
(
g(n)
)
is the set of all functions f (n) for which there are
positive constants c1, c2 and n0 such that
c1g(n) ≤ f (n) ≤ c2g(n) for all n ≥ n0.
c2g(n)
c1g(n)
f (n)
n
n0
g(n) is an asymptotic tight bound for f (n).
CS 270
Algorithms
Oliver
Kullmann
Growth of
Functions
Divide-and-
Conquer
Min-Max-
Problem
Tutorial
Θ-Notation (cont’d)
E.
Distributed solution of stochastic optimal control problem on GPUsPantelis Sopasakis
Stochastic optimal control problems arise in many
applications and are, in principle,
large-scale involving up to millions of decision variables. Their
applicability in control applications is often limited by the
availability of algorithms that can solve them efficiently and within
the sampling time of the controlled system.
In this paper we propose a dual accelerated proximal
gradient algorithm which is amenable to parallelization and
demonstrate that its GPU implementation affords high speed-up
values (with respect to a CPU implementation) and greatly outperforms
well-established commercial optimizers such as Gurobi.
Lagrangian Relaxation And Danzig Wolfe Scheduling Problemmrwalker7
My term project report for my Applied Optimization course. I proposed to examine the application of Lagrangian Relaxation and Danzig-Wolfe Decomposition techniques to the Generalized Assignment Problem. I both presented the formulations and compared the different methods in terms of the solve time metric for cases of varying complexity.
The Generalized Assignment Problem is a mixed-integer problem and a superset of the Scheduling Problem.
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Conference talk at the SIAM Conference on Financial Mathematics and Engineering, held in virtual format, June 1-4 2021, about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model".
- Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
Array is a container which can hold a fix number of items and these items should be of the same type. Most of the data structures make use of arrays to implement their algorithms. Following are the important terms to understand the concept of array.
Algorithm and its Properties
Computational Complexity
TIME COMPLEXITY
SPACE COMPLEXITY
Complexity Analysis and Asymptotic notations.
Big-oh-notation (O)
Omega-notation (Ω)
Theta-notation (Θ)
The Best, Average, and Worst Case Analyses.
COMPLEXITY Analyses EXAMPLES.
Comparing GROWTH RATES
1. Improved Online Scheduling in
Maximizing Throughput of
Equal Length Jobs
Nguyen Kim Thang
(university Paris-Dauphine, France)
presented by Kristoffer Arnsfelt Hansen
(university Aarhus, Denmark)
2. Motivation
Profit maximization
Enterprise: perishable product (electricity, ice-cream, ...).
Clients: single-minded, arrive online, different demands.
Goal: maximize the profit.
Other applications
Data broadcast: online $
broadcast pages
$
ATM network: online packages,
typically of the same length. $
Objective: maximize the total
value.
3. Model
Online Scheduling
Jobs: arrive at ri, processing time
pi, deadline di , value (weight) wi .
Preemption is necessary
Objective: maximize the total
value of jobs completed on time.
Preemption with restart: when a job is scheduled again,
it must be executed from the beginning (e.g., data
broadcast).
Preemption with resume: when a job is scheduled again,
the previously done work can be resumed (e.g., ATM
network) .
5. Competitive ratio
An algorithm ALG is α-competitive if for any instance I
OP T (I)
≤ α (maximization problem)
ALG(I)
What is a competitive ratio?
Measure the performance of an algorithm (worst-case analysis)
The price of an object (the problem):
negotiation
Algorithm Adversary
(upper bound) (lower bound)
6. Contribution
equal processing bounded processing unbounded
times times (by k ) processing times
unit
α=1 α = Θ(log k) ∞
weight
√
3 3
general ≤α≤5 α = Θ(k/ log k) ∞
2
≤ 4.24
7. Contribution
equal processing bounded processing unbounded
times times (by k ) processing times
unit
α=1 α = Θ(log k) ∞
weight
√
3 3
general ≤α≤5 α = Θ(k/ log k) ∞
2
≤ 4.24
Improved algorithms for both models of preemption
Weights and correlation between jobs’ deadlines
10. Starting point
Paradox: low weight, which jobs? higher weight,
imminent deadline later deadline
11. Starting point
Paradox: low weight, which jobs? higher weight,
imminent deadline later deadline
p : initial job length, qj (t) : length of job j at time t
A job j is pending at time t if t + qj (t) ≤ dj
12. Starting point
Paradox: low weight, which jobs? higher weight,
imminent deadline later deadline
p : initial job length, qj (t) : length of job j at time t
A job j is pending at time t if t + qj (t) ≤ dj
A 5-competitive algorithm (preemption with restart)
At any time
If no currently scheduled job, schedule the pending
one with highest weight
If a new pending job arrive with weight at least twice
that of the currently scheduled job, then schedule the
new one (by interrupting the current job)
13. Observations
Correlation among jobs’ deadlines is ignored
Treatment:
A job i is urgent at time t if di < t + qi (t) + p
Some job would be delayed by new urgent jobs
(even with low weight)
Ensure no significant lost if new heavy jobs arrive.
14. Algorithm
Initially, set Q = ∅, α = 0, 1 < β < 3/2
At time t , let i, j be a new released job and the currently scheduled
job, respectively. At any interruption, if α > 0 then α := α + 1
15. Algorithm
Initially, set Q = ∅, α = 0, 1 < β < 3/2
At time t , let i, j be a new released job and the currently scheduled
job, respectively. At any interruption, if α > 0 then α := α + 1
schedule i
If wi ≥ 2wj , wi ≥ 2α w(Q) do
set Q = ∅, α = 0
16. Algorithm
Initially, set Q = ∅, α = 0, 1 < β < 3/2
At time t , let i, j be a new released job and the currently scheduled
job, respectively. At any interruption, if α > 0 then α := α + 1
schedule i
If wi ≥ 2wj , wi ≥ 2α w(Q) do
set Q = ∅, α = 0
schedule job which is
If α = 0, βwj ≤ wi ≤ 2wj do arg max{w : d < t + 2p}
i urgent and dj ≥ t + 2p
set Q = {j}, α = 1
17. Algorithm
Initially, set Q = ∅, α = 0, 1 < β < 3/2
At time t , let i, j be a new released job and the currently scheduled
job, respectively. At any interruption, if α > 0 then α := α + 1
schedule i
If wi ≥ 2wj , wi ≥ 2α w(Q) do
set Q = ∅, α = 0
schedule job which is
If α = 0, βwj ≤ wi ≤ 2wj do arg max{w : d < t + 2p}
i urgent and dj ≥ t + 2p
set Q = {j}, α = 1
If i is urgent do schedule i
wi ≥ 2wj + wj
no job such that
Sj (t) + 2p ≤ d < t + 2p, w ≥ wj
18. The charging scheme
phase of job i
2wf (i)
ALG f (i) i
wj wi
ADV
j i
√
Theorem: the algorithm is (2 + 5)-competitive
√
Theorem: there is a (2 + 5) -competitive algorithm for model
of preemption with resume
19. Conclusion
Improved algorithms for both models of preemption
Open questions:
Settling the right competitive ratio 2.5 ≤ α ≤ 4.24
Interesting: not to reduce the gap but new methods.