Differential conditions for constrained nonlinear programming via Pareto optmization

We deal with differential conditions for local optimality. The conditions that we derive for inequality constrained problems do not require constraint qualifications and are the broadest conditions based on only first-order and second-order derivatives. A similar result is proved for equality constrained problems, although the necessary conditions require the regularity of the equality constraints

Optimal control of the sweeping process over polyhedral controlled sets

The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong $W^{1,2}$-convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples

Global optimality conditions and optimization methods for constrained polynomial programming problems

The general constrained polynomial programming problem (GPP) is considered in this paper. Problem (GPP) has a broad range of applications and is proved to be NP-hard. Necessary global optimality conditions for problem (GPP) are established. Then, a new local optimization method for this problem is proposed by exploiting these necessary global optimality conditions. A global optimization method is proposed for this problem by combining this local optimization method together with an auxiliary function. Some numerical examples are also given to illustrate that these approaches are very efficient. (C) 2015 Elsevier Inc. All rights reserved

Optimal control of ordinary differential equations

3rd cycl

Optimality conditions in continuous-time programming problems

Проблем оптимизације са непрекидним временом састоји се у минимизацији интегралног функционала, са фазним ограничењима различитих типова. Предмет ове докторске дисертације је добијање услова екстремума као и теорема дуалности за класу конвексних и глатких проблема оптимизације са непрекидним вре- меном, са фазним ограничењима типа неједнакости. Нажалост, неки објављени ре- зултати из ове области су нетачни, што је потврђено 2019. године. У раду су добијени нови услови екстремума за поменуту класу проблема. Доказане су теореме слабе и јаке дуалности. Главни апарат за извођење ових резултата је нова тео- рема алтернативе за конвексан систем строгих и нестрогих неједнакости у бесконачно- -димензионим просторима. За примену поменуте теореме, одговарајући услов регу- ларности мора бити задовољен. Неки услови екстремума су изведени уз додатне прет- поставке регуларности ограничења. Теоријски резултати су потврђени практичним примеримаThe continuous-time programming problem consists in minimizing an integral functional, with phase constraints of different types. The subject of this doctoral dissertation is to establish extremum conditions as well as duality theorems for a class of convex and smooth continuous-time programming problems, with phase constraints of the inequality type. Unfortunately, some of the results in this field are not valid, which is confirmed in 2019. In this paper, new optimality conditions for the aforementioned class of problems are ob- tained. The theorems of weak and strong duality are proved. The main tool for deriving these results is a new theorem of the alternative for a convex system of strict and nonstrict inequal- ities in infinite dimensional spaces. In order to apply the aforementioned theorem, a suitable regularity condition must be satisfied. Some optimality conditions are obtained with additional constraint regularity qualification. Theoretical results are confirmed by practical examples

Decomposed Structured Subsets for Semidefinite and Sum-of-Squares Optimization

Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications. Interior-point methods can solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the size of matrix variables and the number of constraints increases. To improve scalability, SDPs can be approximated with lower and upper bounds through the use of structured subsets (e.g., diagonally-dominant and scaled-diagonally dominant matrices). Meanwhile, any underlying sparsity or symmetry structure may be leveraged to form an equivalent SDP with smaller positive semidefinite constraints. In this paper, we present a notion of decomposed structured subsets}to approximate an SDP with structured subsets after an equivalent conversion. The lower/upper bounds found by approximation after conversion become tighter than the bounds obtained by approximating the original SDP directly. We apply decomposed structured subsets to semidefinite and sum-of-squares optimization problems with examples of H-infinity norm estimation and constrained polynomial optimization. An existing basis pursuit method is adapted into this framework to iteratively refine bounds.Comment: 23 pages, 10 figures, 9 table

Function space methods in optimal control with applications to power systems

Imperial Users onl

Group-invariant Semidefinite Programming and Applications

Abstract This essay considers semidefinite programming problems that exhibit a special form of symmetry called group-invariance. We demonstrate the effect of such symmetries on certain path-following interior-point algorithms, and highlight a reduction technique that is particularly useful on certain groupinvariant semidefinite programming problems. Two applications of groupinvariant semidefinite programming problems-one in truss design and the other in graph theory-are presented. ii Acknowledgements To my supervisor, Dr. Chek Beng Chua, for his wise advice, great encouragement and continuous support during my master's study. To Professor Michael Best for his comments and careful reading of the draft. To my dear friends who gave me great help and made my life in Waterloo a wonderful experience

Algebraic Algorithm Design and Local Search

Formal, mathematically-based techniques promise to play an expanding role in the development and maintenance of the software on which our technological society depends. Algebraic techniques have been applied successfully to algorithm synthesis by the use of algorithm theories and design tactics, an approach pioneered in the Kestrel Interactive Development System (KIDS). An algorithm theory formally characterizes the essential components of a family of algorithms. A design tactic is a specialized procedure for recognizing in a problem specification the structures identified in an algorithm theory and then synthesizing a program. Design tactics are hard to write, however, and much of the knowledge they use is encoded procedurally in idiosyncratic ways. Algebraic methods promise a way to represent algorithm design knowledge declaratively and uniformly. We describe a general method for performing algorithm design that is more purely algebraic than that of KIDS. This method is then applied to local search. Local search is a large and diverse class of algorithms applicable to a wide range of problems; it is both intrinsically important and representative of algorithm design as a whole. A general theory of local search is formalized to describe the basic properties common to all local search algorithms, and applied to several variants of hill climbing and simulated annealing. The general theory is then specialized to describe some more advanced local search techniques, namely tabu search and the Kernighan-Lin heuristic