course work of optimization theory and algorithms

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• Prof. James Orlin
• Dr. Ebrahim Nasrabadi

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• Sloan School of Management

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• Operations Management
• Project Management
• Systems Optimization
• Applied Mathematics
• Probability and Statistics
• Game Theory

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Optimization methods in management science, course description.

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AM221. Advanced Optimization

This is a graduate level course on optimization which provides a foundation for applications such as statistical machine learning, signal processing, finance, and approximation algorithms. The course will cover fundamental concepts in optimization theory, modeling, and algorithmic techniques for solving large-scale optimization problems. Topics include elements of convex analysis, linear programming, Lagrangian duality, optimality conditions, and discrete and combinatorial optimization. Exercises and the class project will involve developing and implementing optimization algorithms. http://rasmuskyng.com/am221_spring18/ See also: Courses , Spring 2018 , AM

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Optimization Theory and Algorithms

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Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

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ECE 592 611 Optimizations and Algorithms

3 Credit Hours

(also offered as  CSC 591 ) This course introduces advances in optimization theory and algorithms with rapidly growing applications in machine learning, systems, and control. Methods to obtain the extremum (minimum or maximum) of a non-dynamic system and the use of these methods in various engineering applications are given.

Prerequisite

Introductory courses in probability and linear algebra.

Course Objectives

• The goal of this course is to prepare graduate students with a solid theoretical and mathematical foundation and applied techniques at the intersection of optimization, algorithms, and machine learning to conduct advanced research in the related fields.
• Students will gain expertise in designing algorithms based on conventional techniques and be able to deal with intractable problems and implement algorithms given the description.
• Students will need to undertake a half-semester long project that practices the optimization theory and algorithms in their areas of interest. It is allowed to be a replication or an improvement of a known solving strategy for a given optimization problem to assess/compare performance characteristics.

Topics covered

This course aims to cover the following topics:

• Nonlinear unconstrained optimization, linear programming, nonlinear constrained optimization, computational and search methods for optimization; convex optimization and integer programming.
• Greedy, divide and conquer, dynamic programming; approximation algorithms.
• Stochastic optimization, sparsity, regularized optimization, interior-point methods, proximal methods, robust optimization.
• Convergence rate analysis, momentum-based acceleration, distributed and asynchronous algorithm design, saddle point escaping.

Course Status :	Upcoming
Course Type :	Elective
Duration :	12 weeks
Category :
Credit Points :	3
Undergraduate/Postgraduate
Start Date :	22 Jul 2024
End Date :	11 Oct 2024
Enrollment Ends :	29 Jul 2024
Exam Registration Ends :	16 Aug 2024
Exam Date :	27 Oct 2024 IST

Homework	20%
Midterm Exam	25%
Final Exam	30%
Project	25%

E.K.P. Chong and S.H. Zak, “An Introduction to Optimization,” John Wiley & Sons, 2008.

A list of essential and trending papers will be provided on the course website. Useful reference books on optimization theory and mathematical backgrounds include:

S. Boyd and L. Vandenberghe, “Convex Optimization,” Cambridge University Press, 2004.

Y. Nesterov, “Introductory Lectures on Convex Optimization: A Basic Course,” Springer, 2004.

M. Bazarra, H.D. Sherali, and C.M. Shetty, “Nonlinear Programming: Theory and Algorithms,” John Wiley & Sons, 2006.

Updated 7/8/2020

UC Davis TETRAPODS Institute of Data Science

Optimization theory and algorithms.

Theme II: Optimization Theory and Algorithms for Machine Learning including Numerical Solvers for Large-Scale Nonconvex and Nonsmooth Learning Problems

Participants:

• CS - P. Devanbu, I. Tagkopoulos
• ECE - L. Lai
• Math - J. De Loera, A. Fannjiang, M. Köppe, S. Ma, T. Strohmer
• Stat - K. Balasubramanian, X. Li

Algorithmic tools from optimization have become more and more important in machine learning and data analysis. Increasingly, the focus is shifting to nonconvex optimization, in which the functions to be minimized may have a plethora of local solutions and saddle points. Nonconvex problems appear in deep learning, as well as in important applications in signal and image processing. Examples include image classification, data clustering, and phase retrieval. Another important direction is how to implement nonconvex methods on devices with very limited resources. Solving this challenge is key for on-device machine learning which offers great benefits such as increased privacy and security .

UCD4IDS faculty have played key roles in the recent surge of nonconvex optimization. A very fruitful direction has been to find a proper convex relaxation of the nonconvex problem and demonstrate that under practically relevant conditions the solution of the convex problem coincides with that of the original nonconvex one. The now well-known example is the PhaseLift approach to solve the famous phase retrieval problem. In turn, we kicked off the theoretical analysis of nonconvex geometry for signal processing and machine learning, while our more recent work represents a milestone in blind deconvolution , thereby igniting extensive work in biconvex optimization.

Our research projects in this theme include:

• Stochastic algorithms (improving Stochastic Gradient Descent algorithm)
• Optimization landscape of nonconvex problems including analysis on escape from a saddle point
• Privacy and security in machine learning
• Applications-Phase retrieval and beyond

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Optimization Theory and Algorithms

Course detail.

Description

This course covers basic theory and algorithms for unconstrained and constrained optimization problems, convex and non-convex optimization problems, optimality conditions including duality theory. Algorithms include basic first-order and second-order methods. Some applications of optimization, such as those in data science, will be introduced. The course also requires algorithm implementation and problem solving on computers.

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Second-Order Variational Analysis in Optimization, Variational Stability, and Control

Theory, Algorithms, Applications

• © 2024
• Boris S. Mordukhovich 0

Department of Mathematics, Wayne State University Department of Mathematics, Detroit, USA

You can also search for this author in PubMed   Google Scholar

• Copious exercises at various levels of difficulties with solution hints for more challenging ones
• Open questions and conjectures with extended commentaries
• Rapidly developing area of second-order variational analysis and its applications

Part of the book series: Springer Series in Operations Research and Financial Engineering (ORFE)

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Table of contents (9 chapters)

Front matter, basic concepts of second-order analysis.

Boris S. Mordukhovich

Second-Order Subdifferential Calculus

Evaluating second-order subdifferentials, lipschitzian stability via second-order subdifferentials, full stability of local minimizers, full stability in variational systems, full stability in pde optimal control, variational convexity in optimization, second-order numerical variational analysis, back matter.

• cone mappings
• generalized Newton method
• subdifferential calculus
• parametric variational systems
• elliptic PDEs

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Book Title : Second-Order Variational Analysis in Optimization, Variational Stability, and Control

Book Subtitle : Theory, Algorithms, Applications

Authors : Boris S. Mordukhovich

Series Title : Springer Series in Operations Research and Financial Engineering

DOI : https://doi.org/10.1007/978-3-031-53476-8

Publisher : Springer Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : he Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024

Hardcover ISBN : 978-3-031-53475-1 Published: 22 May 2024

Softcover ISBN : 978-3-031-53478-2 Due: 09 July 2024

eBook ISBN : 978-3-031-53476-8 Published: 21 May 2024

Series ISSN : 1431-8598

Series E-ISSN : 2197-1773

Edition Number : 1

Number of Pages : XVIII, 789

Number of Illustrations : 3 b/w illustrations, 21 illustrations in colour

Topics : Optimization , Analysis , Operations Research, Management Science

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Fall 2022

Optimization Theory and Algorithm II / Fall 2022

• New Lecture is up: Lecture 14 Course Review
• New Lecture is up: Lecture 13 Alternating Deirection Method of Multipliers [ notes ]
• New Assignment released: [ HW4 ]
• New Lecture is up: Lecture 12 Block Coordinate Descent [ notes ]
• New Lecture is up: Lecture 11 Federated Optimization [ notes ]
• New Lecture is up: Lecture 10 Stochastic Variance Reduced Gradient [ notes ]
• New Assignment released: [ HW3 ]

Course Description

This course is to present the basic theories and algorithms in optimization fields for undergraduate students at School of Management, Xi'an Jiaotong University. The aim is to give students a thorough understanding of how to constructe algorithms for solving optimization problems, and how to develop theories to analyze optimization problems and corresponding algorithms.

Optimization is a special field that is built on three interwined pillars:

• Models give rise to optimization problems.
• Algorithms solve optimization problems.
• Theoretical foundations support algorithms and models.

Optimization = Modeling + Algorithm + Theory

Location: B103 Main Building (主楼B103)

Date : 1st&2nd class, Mondays and 9&10th class, Thursdays, 09/05 - 10/24, 2022.

Instructors

Xiangyu Chang

Teaching Assistants

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Computer Science > Computer Science and Game Theory

Title: second-order algorithms for finding local nash equilibria in zero-sum games.

Abstract: Zero-sum games arise in a wide variety of problems, including robust optimization and adversarial learning. However, algorithms deployed for finding a local Nash equilibrium in these games often converge to non-Nash stationary points. This highlights a key challenge: for any algorithm, the stability properties of its underlying dynamical system can cause non-Nash points to be potential attractors. To overcome this challenge, algorithms must account for subtleties involving the curvatures of players' costs. To this end, we leverage dynamical system theory and develop a second-order algorithm for finding a local Nash equilibrium in the smooth, possibly nonconvex-nonconcave, zero-sum game setting. First, we prove that this novel method guarantees convergence to only local Nash equilibria with a local linear convergence rate. We then interpret a version of this method as a modified Gauss-Newton algorithm with local superlinear convergence to the neighborhood of a point that satisfies first-order local Nash equilibrium conditions. In comparison, current related state-of-the-art methods do not offer convergence rate guarantees. Furthermore, we show that this approach naturally generalizes to settings with convex and potentially coupled constraints while retaining earlier guarantees of convergence to only local (generalized) Nash equilibria.

Subjects:	Computer Science and Game Theory (cs.GT); Multiagent Systems (cs.MA); Systems and Control (eess.SY)
Cite as:	[cs.GT]
	(or [cs.GT] for this version)
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