Alexander Strekalovsky

• Institute for System Dynamics and Control Theory, Russian Academy of Sciences

This book constitutes refereed proceedings of the 20th International Conference on Mathematical Optimization Theory and Operations Research, MOTOR 2021, held in Irkutsk, Russia, in July 2021. Due to the COVID-19 pandemic the conference was held online. The 31 full papers and 3 short papers presented in this volume were carefully reviewed and selec...

This chapter addresses a new methodology for finding optimistic solutions in bilevel optimization problems (BOPs). In Introduction, we present our view of the classification for corresponding numerical methods available in the literature. Then we focus on the quadratic case and describe the reduction of BOPs with quadratic objective functions to on...

This paper addresses the general optimization problem (\(\mathcal P\)) with equality and inequality constraints and the cost function given by d.c. functions. We reduce the problem to a penalized problem (\(\mathcal P_{\sigma }\)) without constraints with the help of the Exact Penalization Theory. Further, we show that the reduced problem is also a...

This paper addresses the numerical solution of fractional programs with quadratic functions in the ratios. Instead of considering a sum-of-ratios problem directly, we developed an efficient global search algorithm, which is based on two approaches to the problem. The first one adopts a reduction of the fractional minimization problem to the solutio...

The chapter addresses the nonsmooth optimization problem with the objective function and equality and inequality constraints given by DC functions. First, the original problem is reduced to a problem without constraints by the exact penalization theory, so that the reduced (penalized) problem is also a DC minimization problem. Then, we develop a lo...

This paper addresses the nonconvex optimization problem with the cost function and equality and inequality constraints given by d.c. functions. The original problem is reduced to a problem without constraints by means of the exact penalization techniques. Furthermore, the penalized problem is presented as a d.c. minimization problem. For the latter...

This paper addresses the nonconvex optimization problem with the cost function and the inequality constraints given by d.c. functions. The original problem is reduced to a problem without inequality constraints by means of the exact penalization techniques. Furthermore, the penalized problem is presented as a d.c. minimization problem. For the latt...

The paper addresses the nonconvex nonsmooth optimization problem with the cost function and equality and inequality constraints given by d.c. functions. The original problem is reduced to a problem without constraints with the help of the exact penalization theory. After that, the penalized problem is represented as a d.c. minimization problem with...

This paper addresses a rather general fractional optimization problem. There are two ways to reduce the original problem. The first one is a solution of an equation with the optimal value of an auxiliary d.c. optimization problem with a vector parameter. The second one is to solve the second auxiliary problem with nonlinear inequality constraints....

This paper addresses a nonconvex optimization problem with the cost function and inequality constraints given by d.c. functions. The original problem is reduced to a problem without inequality constraints by the exact penalization procedure. A special local search method for the penalized problem is developed, which is based, first, on the lineariz...

This paper addresses the development of efficient global search methods for fractional programming problems. Such problems are, in general, nonconvex (with numerous local extremums) and belong to a class of global optimization problems. First, we reduce a rather general fractional programming problem with d.c. functions to solving an equation with...

In this paper, we address the nonconvex optimization problem, with the goal function and the inequality constraints given by the functions represented by the difference of convex functions. The effectiveness of the classical Lagrange function and the max-merit function is being investigated as the merit functions of the original problem. In additio...

We consider the problem of optimizing the sum of several rational functions via reduction to a problem with d.c. constraints. We propose a method of finding a local solution to the fractional program which can be subsequently used in the global search method based on the global optimality conditions for a problem with nonconvex (d.c.) constraints [...

This paper addresses a rather general problem of nonlinear optimization with the inequality constraints and the goal function defined by the (d.c.) functions represented by the difference of two convex functions. In order to reduce the constrained optimization problem to an unconstrained one, we investigate three auxiliary problems with the max-mer...

We generalize Malfatti’s problem which dates back to 200 years ago as a global optimization problem in a high dimensional space. The problem has been formulated as the convex maximization problem over a nonconvex set. Global optimality condition by Strekalovsky[11] has been applied to this problem. For solving numerically Malfatti’s problem, we pro...

Optimal control problems with a nonconvex quadratic functional of Lagrange are considered. On the base of global optimality conditions we develop a global search algorithm, one of the principal module of which is represented by special local search method. The results of computational testing presented.

Using the piecewise-linear function, consideration was given to the problem of separation of the sets whose convex hulls have nonempty intersections. For the problem of polyhedral separability, an algorithm to solve the equivalent optimization problem of seeking the family of separating hyperplanes was proposed and substantiated. Its efficiency was...

It is well known that short run cost functions of firms are convex functions when production functions are concave [14]. Average cost minimiza- tion as a classical economics problem has been studied in fundamental text- books [14, 4, 7, 8] and in the literature [2, 3, 9, 12, 13, 1]. However, it seems that less attention so far has been paid to the...

The separation problem of two sets, whose convex hulls have a nonempty intersection, is considered. In order to find a solution of the problem algorithms of local and global search are developed. The efficiency of the algorithms is demonstrated by computational simulations on test examples.

First, we consider a d.c. minimization problem with a simple feasible set and develop a special method based on the linearization with respect to the basic nonconvexity. The convergence of the methods is analyzed and compared with published results. Theoretical and practical stopping criteria are proposed. Second, we consider a problem with d.c. co...

The problem of numerical finding of a Nash equilibrium in a 3-player polymatrix game is considered. Such a game can be completely described by six matrices, and it turns out to be equivalent to the solving a nonconvex optimization problem with a bilinear structure in the objective function. Special methods of local and global search for the optimiz...

Here we consider three very popular optimization problems: the linear complementarity problem, the search for Nash equilibria in a bimatrix game, and the quadratic-linear bilevel programming problem. It can be shown that each of the problem possesses a hidden nonconvexity and, as a consequence, a rather large number of local solutions which are dif...

The Nonconvex Optimal Control Problem with functions represented by the difference of two convex functions in terminal and integrand parts is considered. Global optimality conditions, bound up with the Pontryagin maximum principle, are proved, discussed, and illustrated by examples.

A nonconvex optimal control problem with the Bolza objective functional is considered. The objective functional is specified by functions represented by the difference of two convex functions (Alexandrov functions). New necessary and sufficient global optimality conditions for minimizing sequences of controls are proved. These conditions provide th...

We consider the maximization problem for an integral functional with a state-convex integrand function along a standard control system. We show necessary and sufficient global optimality conditions related to the Pontryagin’s maximum principle. We study the properties of these conditions and their relations with optimal control theory. We also illu...

The linear-linear and quadratic-linear bilevel programming problems are considered. Their optimistic statement is reduced to a nonconvex mathematical programming problem with the bilinear structure. Approximate algorithms of local and global search in the obtained problems are proposed. The results of computational solving randomly generated test p...

A quadratic-linear bilevel programming problem is considered. Its optimistic statement is reduced to a series of nonconvex unilevel problems. An approximate algorithm for global search in reduced problems is proposed. Numerical solutions of randomly generated test problems are given and analyzed. Key wordsbilevel programming-optimistic solution-no...

A quadratic-linear bilevel programming problem is considered. Its optimistic statement is reduced to a nonconvex mathematical programming problem with a quadratic-bilinear structure. An approximate algorithm of a local search in the problem obtained is proposed, proved, and tested. Key wordsbilevel programming-optimistic solution-nonconvex optimiz...

The well-known linear complementarity problem with definite matrices is considered. It is proposed to solve it using a global optimization algorithm in which one of the basic stages is a special local search. The proposed global search algorithm is tested using a variety of randomly generated problems; a detailed analysis of the computational exper...

A nonconvex optimal control problem is examined for a system that is linear with respect to state and has a terminal objective functional representable as the difference of two convex functions. A new local search method is proposed, and its convergence is proved. A strategy is also developed for the search of a globally optimal control process, be...

Nonconvex optimization problems with an inequality constraint given by the difference of two convex functions (by a d.c. function) are considered. Two methods for finding local solutions to this problem are proposed that combine the solution of partially linearized problems and descent to a level surface of the d.c. function. The convergence of the...

Two control problems for a state-linear control system are considered: the minimization of a terminal functional representable as the difference of two convex functions (d.c. functions) and the minimization of a convex terminal functional with a d.c. terminal inequality contraint. Necessary and sufficient global optimality conditions are proved for...

In this paper we propose two variants of Local Search Method for reverse convex problems. The first is based on well-known theorem of H. Tuy as well as on Linearization Principle. The second variant is due to an idea of J. Rosen. We demonstrate the practical effectiveness of the proposed methods computationally.

Nonconvex optimization problems with a single inequality constraint given by the difference of two convex functions (i.e., by a d.c. function) are considered. Such problems may have many local solutions and stationary points that are far (in terms of, say, the value of the objective function) from a global solution. Necessary and sufficient conditi...

On the base of global optimality conditions for RCP we develop the global search strategy (GSS) and focus on the global convergence of GSS giving a variant of the proof.

A new approach to the solution of nonconvex problems of optimal control and mathematical programming is treated, which rests on the theory of global optimality conditions (GOC). Moreover, attention is given to the development and investigation of special methods of local search and to investigation of the convergence of strategies of global search,...

The Maximum Clique Problem (MCP) is regarded here as the maximization of an indefinite quadratic form over the canonical simplex. For solving MCP an algorithm based upon Global Optimality Conditions (GOC) is applied. Furthermore, each step of the algorithm is analytically investigated and tested. The computational results for the proposed algorithm...

We give necessary and sufficient Global Optimality Conditions for d.c. (difference of two convex functions) minimization problem and further development of a global search algorithm based on the theory. Finally we test the proposed algorithm on a concret problem.

In this paper we give an analytical equivalent for the inclusion of a set to the Lebesque set of a convex function. Using this results, we obtain global optimality conditions (GOC) related to classical optimization theory for convex maximization and reverse-convex optimization. Several examples illustrate the effectiveness of these optimality condi...

In this paper we consider non-convex optimal control problems having the same goal: to maximize a convex function of the terminal state. A global search algorithm is given. The first numerical tests have been performed.

We consider two kinds of nonconvex problems: convex maximization and reverseconvex optimization. Using the new information about the problems in the form of Global Optimality Search Algorithms [1–5], we construct Global Search Algorithms and study their global convergence. Numerical experiments also presented here are rather promising especially fo...

A nonconvex optimal control problem whose nonconvexity is generated by an integro-terminal objective functional is considered. A new local search method that allows obtaining a control process x * (·), u * (·) satisfying, in particular, Pontryagin's maximum principle is proposed. Some peculiar properties of convergence of the algorithm are studied....

In this paper we propose a new approach based on global optimality conditions for solving continuous nonconvex optimization problems. We present in detail a technique for nding a solution to the following three problems: the problem of polyhedral separability, the problem of solving a system of nonlinear equations, and the problem of the Nash equil...