Optimization Methods 2022 - История изменений

NATab в 07:55, 14 февраля 2022

2022-02-14T07:55:20Z

← Предыдущая версия		Версия от 07:55, 14 февраля 2022
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	- Optimization problems with linear equality constraints: first and second-order optimality conditions, Lagrange function.		- Optimization problems with linear equality constraints: first and second-order optimality conditions, Lagrange function.

	- General smooth optimization problems with equality and inequality constraints: first and second-order optimality conditions, dual problems,		- General smooth optimization problems with equality and inequality constraints: first and second-order optimality conditions, dual problems, penalty and augmented Lagrangian methods, sequential quadratic programming.
	penalty and augmented Lagrangian methods, sequential quadratic programming.

	- Convex optimization: optimality conditions, duality, subgradients and subdifferential, cutting planes and bundle methods, the complexity of convex optimization.		- Convex optimization: optimality conditions, duality, subgradients and subdifferential, cutting planes and bundle methods, the complexity of convex optimization.
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	- Linear programming: duality, decomposition.		- Linear programming: duality, decomposition.

	- Interior-point methods.		- Interior-point methods.



	= Recommended Bibliography =		= Recommended Bibliography =

NATab в 07:55, 14 февраля 2022

2022-02-14T07:55:01Z

NATab: Новая страница: «Abstract The course gives a comprehansive foundation for theory, methods and algorithms of mathematical optimization. The prerequisites are linear algebra and c…»

2022-02-14T07:47:56Z

Новая страница: «Abstract The course gives a comprehansive foundation for theory, methods and algorithms of mathematical optimization. The prerequisites are linear algebra and c…»

Новая страница

Abstract

The course gives a comprehansive foundation for theory, methods and algorithms of mathematical optimization. The prerequisites are
linear algebra and calculus.

Learning Objectives

- Students will study main concepts of optimization theory and develop methodology for theoretical investigaion of optimization problems.

- Students will obtain an understanding of creation, effectiveness and application optimization methods and algorithms on practice.

- The course will give students the possibility of solving standadrd and nonstandard mathematical problems connected to finding optimal solutions.

Expected Lerning Outcomes

- Students should be able to classify optimization problems according to their mathematical properties.

- Students should be able to perform a theoretical investigation of a given optimization problem in order to assess its complexity.

- Students should be able to write down first and second order optimality cobditions.

- Students should be able to provide a dual analysis of linear and convex optimization problems.

- Students should be able to asses the rate of convergence of the first and second order optimization methods.

- Students should be able to solve simple optimization problems without computer.

- Students should be able to implement different optimization codes in a computer enviroment.

- Students should be able to analyse the obtained solutions.

Course content

- Unvariate optimization: unimodal (invex) functions, golden section method, Lipschitzian optimization.

- Unconstrained quadratic optimization: algebraic approach, complete characterization of stationary points, gradient method with exact line search, conjugate gradient method.

- General unconstrained optimization: first and second order optimality conditions, descent directions, steepest descent method, conjugate gradient methods, Newton method, quasi-Newton methods, inexact line search, rate of convergence.

- Optimization problems with linear equality constraints: first and second order optimality conditions, Lagrange function.

- General smooth optimization problems with equality and inequality constraints: first and second order optimality conditions, dual problems,
penalty and augmented Lagrangian methods, sequantial quadratic programming.

- Convex optimization: optimality conditions, duality, subgradients and subdifferential, cutting planes and bundle methods, complexity of convex optimization.

- Linear programming: duality, decomposition.

- Interior-point methods.

Recommended Bibliography

- Nocedal J., Wright S.J. Numerical optimization. Second Edition. Springer, 2006

- Bazaraa M. S., Sherali H.D, Shetty C. M. Nonlinear Programming: Theory and Algorithms 3rd Edition, John Wiley & Sons, 2006

- Beck, A. First-Order Methods in Optimization, MOS-SIAM Series on Optimization, 2017

- Ben-Tal A., Arkadi Nemirovski A. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MOS-SIAM Series on Optimization, 2001

- Nesterov Yu. Introductory Lectures on Convex Optimization, Springer US, 2004

← Предыдущая версия		Версия от 07:55, 14 февраля 2022
Строка 1:		Строка 1:
	Abstract		= Abstract =

	The course gives a ~~comprehansive~~ foundation for theory, methods and algorithms of mathematical optimization. The prerequisites are		The course gives a comprehensive foundation for theory, methods and algorithms of mathematical optimization. The prerequisites are
	linear algebra and calculus.		linear algebra and calculus.

	Learning Objectives		= Learning Objectives =

	- Students will study main concepts of optimization theory and develop methodology for theoretical ~~investigaion~~ of optimization problems.		- Students will study the main concepts of optimization theory and develop a methodology for theoretical investigation of optimization problems.

	- Students will obtain an understanding of creation, effectiveness and application optimization methods and algorithms on practice.		- Students will obtain an understanding of creation, effectiveness and application optimization methods and algorithms on practice.

	- The course will give students the possibility of solving ~~standadrd~~ and nonstandard mathematical problems connected to finding optimal solutions.		- The course will give students the possibility of solving standard and nonstandard mathematical problems connected to finding optimal solutions.

	Expected ~~Lerning~~ Outcomes		= Expected Learning Outcomes =

	- Students should be able to classify optimization problems according to their mathematical properties.		- Students should be able to classify optimization problems according to their mathematical properties.
Строка 18:		Строка 18:
	- Students should be able to perform a theoretical investigation of a given optimization problem in order to assess its complexity.		- Students should be able to perform a theoretical investigation of a given optimization problem in order to assess its complexity.

	- Students should be able to write down first and second order optimality ~~cobditions~~.		- Students should be able to write down first and second-order optimality conditions.

	- Students should be able to provide a dual analysis of linear and convex optimization problems.		- Students should be able to provide a dual analysis of linear and convex optimization problems.

	- Students should be able to ~~asses~~ the rate of convergence of the first and second order optimization methods.		- Students should be able to assess the rate of convergence of the first and second-order optimization methods.

	- Students should be able to solve simple optimization problems without computer.		- Students should be able to solve simple optimization problems without a computer.

	- Students should be able to implement different optimization codes in a computer ~~enviroment~~.		- Students should be able to implement different optimization codes in a computer environment.

	- Students should be able to analyse the obtained solutions.		- Students should be able to analyse the obtained solutions.


	Course content		= Course content =

	- Unvariate optimization: unimodal (invex) functions, golden section method, Lipschitzian optimization.		- Unvariate optimization: unimodal (invex) functions, golden section method, Lipschitzian optimization.

	- Unconstrained quadratic optimization: algebraic approach, complete characterization of stationary points, gradient method with exact line search, conjugate gradient method.		- Unconstrained quadratic optimization: algebraic approach, complete characterization of stationary points, gradient method with exact line search, conjugate gradient method.

	- General unconstrained optimization: first and second order optimality conditions, descent directions, steepest descent method, conjugate gradient methods, Newton method, quasi-Newton methods, inexact line search, rate of convergence.		- General unconstrained optimization: first and second-order optimality conditions, descent directions, steepest descent method, conjugate gradient methods, Newton method, quasi-Newton methods, inexact line search, rate of convergence.

	- Optimization problems with linear equality constraints: first and second order optimality conditions, Lagrange function.		- Optimization problems with linear equality constraints: first and second-order optimality conditions, Lagrange function.

	- General smooth optimization problems with equality and inequality constraints: first and second order optimality conditions, dual problems,		- General smooth optimization problems with equality and inequality constraints: first and second-order optimality conditions, dual problems,
	penalty and augmented Lagrangian methods, ~~sequantial~~ quadratic programming.		penalty and augmented Lagrangian methods, sequential quadratic programming.

	- Convex optimization: optimality conditions, duality, subgradients and subdifferential, cutting planes and bundle methods, complexity of convex optimization.		- Convex optimization: optimality conditions, duality, subgradients and subdifferential, cutting planes and bundle methods, the complexity of convex optimization.

	- Linear programming: duality, decomposition.		- Linear programming: duality, decomposition.
Строка 52:		Строка 52:


	Recommended Bibliography		= Recommended Bibliography =

	- Nocedal J., Wright S.J. Numerical optimization. Second Edition. Springer, 2006		- Nocedal J., Wright S.J. Numerical optimization. Second Edition. Springer, 2006