Ider Tseveendorj

Optimization and optimal control

Extragradient Approach to the Solution of Two Person Non-Zero Sum Games (A Antipin) Nonlinear Phenomena in Economics (S Budnyam) On Some Theory, Methods and Algorithms for Concave Programming (R Enkhbat) Maximum Clique Regularizations (D Fortin & I Tseveendorj) The Role of Optimization in Economics (D W Katzner) A Global Optimization Approach to Solving Equilibrium Programming Problems (O Khamisov) Comparison of Convex Relaxations for Monomials of Odd Degree (L Liberti) Structural Stability of Vector Optimization Problems (P Mbunga) Optimization for Black-Box Objective Functions (H Nakayama et al.) New Variants of Gradient Type Methods in Optimal Control Problems (V A Srochko) Controlled Systems with Distributed Parameters: Optimally Conditions and Optimization Methods in Class of Smooth Restricted Controls (O V Vasiliev) The Points of a Linear Manifold Nearest to a Given Vector (V I Zorkaltsev) and other articles.

Piecewise convex maximization approach to multiknapsack

We refine the reverse convex approach to binary programs into a piecewise convex maximization problem with only two pieces. As a vital lead, we address the multiknapsack problem. The results of computational experiments are presented in contrast with the best known solutions found by heuristics and by the reverse convex approach.

Piece adding technique for convex maximization problems

In this article we provide an algorithm, where to escape from a local maximum y of convex function f over D, we (locally) solve piecewise convex maximization max{min{f (x) − f (y), py(x)} | x ∈ D} with an additional convex function py(·). The last problem can be seen as a strictly convex improvement of the standard cutting plane technique for convex maximization. We report some computational results, that show the algorithm efficiency.

Piecewise Convex Maximization Problems: Piece Adding Technique

In this article, we provide a global search algorithm for maximizing a piecewise convex function F over a compact D. We propose to iteratively refine the function F at local solution y by a virtual cutting function py(⋅) and to solve max {min {F(x)−F(y),py(x)}∣x∈D} instead. We call this function either a patch, when it avoids returning back to the same local solutions, or a pseudo patch, when it possibly yields a better point. It is virtual in the sense that the role of cutting constraints is played by additional convex pieces in the objective function. We report some computational results, that represent an improvement on previous linearization based techniques.

Q-subdifferential and Q-conjugate for global optimality

Normal cone and subdifferential have been generalized through various continuous functions; in this article, we focus on a non separable Q-subdifferential version. Necessary and sufficient optimality conditions for unconstrained nonconvex problems are revisited accordingly. For inequality constrained problems, Q-subdifferential and the lagrangian multipliers, enhanced as continuous functions instead of scalars, allow us to derive new necessary and sufficient optimality conditions. In the same way, the Legendre-Fenchel conjugate is generalized into Q-conjugate and global optimality conditions are derived by Q-conjugate as well, leading to a tighter inequality.

Generalized subdifferentials of the sign change counting function

The counting function on binary values is extended to the signed case in order to count the number of transitions between contiguous locations. A generalized subdifferential for this sign change counting function is given where classical subdifferentials remain intractable. An attempt to prove global optimality at some point, for the 4-dimensional first non trivial example, is made by using a sufficient condition specially tailored among all the cases for this subdifferential.

Attractive force search algorithm for piecewise convex maximization problems

We consider mathematical programming problems with the so-called piecewise convex objective functions. A solution method for this interesting and important class of nonconvex problems is presented. This method is based on Newton’s law of universal gravitation, multicriteria optimization and Helly’s theorem on convex bodies. Numerical experiments using well known classes of test problems on piecewise convex maximization, convex maximization as well as the maximum clique problem show the efficiency of the approach.

A trust branching path heuristic for zero-one programming

For 0-1 problems, we propose an exact Branch and Bound procedure where branching strategy is based on empirical distribution of each variable within three intervals [0,[epsilon]],[[epsilon],1-[epsilon]],[1-[epsilon],1] under the linear relaxation model. We compare the strategy on multiknapsack and maximum clique problems with other heuristics.

Survey of Piecewise Convex Maximization and PCMP over Spherical Sets

The main investigation in this chapter is concerned with a piecewise convex function which can be defined by the pointwise minimum of convex functions, \(F(x) =\min \{ f_{1}(x),\ldots,f_{m}(x)\}\). Such piecewise convex functions closely approximate nonconvex functions, that seems to us as a natural extension of the piecewise affine approximation from convex analysis. Maximizing F(⋅ ) over a convex domain have been investigated during the last decade by carrying tools based mostly on linearization and affine separation. In this chapter, we present a brief overview of optimality conditions, methods, and some attempts to solve this difficult nonconvex optimization problem. We also review how the line search paradigm leads to a radius search paradigm, in the sense that sphere separation which seems to us more appropriate than the affine separation. Some simple, but illustrative, examples showing the issues in searching for a global solution are given.

Minimizing Sign Changes Rowwise: Consecutive Ones Property and Beyond

A 0–1 matrix where in each row the 1s occur consecutively is said to have the consecutive 1s property. Since, this property is scarcely fulfilled in real problems and since it is non-deterministic polynomial time (NP)-hard to find the nearest arrangement to the property, we give a quadratic assignment formulation for optimizing the distance to the property. The formulation carries over the sign case with\(0,+1,-1\) matrix entries. We discuss and compare this exact approach, for both signed and unsigned cases, with spectral approaches based on bisection instead.