The discretization approach for solving semi-infinite optimization problems is considered. We are interested in the rate of the approximation error between the solution of the semi-infinite problem and the solution of the discretized program depending on the discretization mesh-size d. It will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether t...

In Machine Learning (ML) algorithms, one of the crucial issues is the representation of the data. As the data become heterogeneous and large-scale, single kernel methods become insufficient to classify nonlinear data. The finite combinations of kernels are limited up to a finite choice. In order to overcome this discrepancy, we propose a novel method of ”infinite” kernel combinations for learni...

In this paper we discuss duality theory of optimization problems with a linear objective function and subject to linear constraints with cone inclusions, referred to as conic linear problems. We formulate the Lagrangian dual of a conic linear problem and survey some results based on the conjugate duality approach where the questions of “no duality gap” and existence of optimal solutions are rel...

We consider the class of semi-infinite programming problems which became in recent years a powerful tool for the mathematical modelling of many real-life problems. In this paper, we study an augmented Lagrangian approach to semi-infinite problems and present necessary and sufficient conditions for the existence of corresponding augmented Lagrange multipliers. Furthermore, we discuss two particu...

We introduce a “generalized small inverse problem (GSIP)” and present an algorithm for solving this problem. GSIP is formulated as finding small solutions of f(x0, x1, . . . , xn) = x0h(x1, . . . , xn) + C = 0(mod M) for an n-variate polynomial h, non-zero integers C and M . Our algorithm is based on lattice-based Coppersmith technique. We provide a strategy for construction of a lattice basis ...

Abstract. An infinite-dimensional linear programming formulated on L1 spaces, problem (P), is studied in this paper. A related optimization problem, general capacity problem (GCAP), is also mentioned in this paper. But we find that the optimal solution does not exist in problem (P). Thus, we approach the optimal value for problem (P) via solving the problem (GCAP). A proposed algorithm is shown...

Motivated by important applications, the theory of mathematical programming has been extended to the case of infinitely many restrictions. At the same time, this theory knew remarcable developments since invexity and its further generalizations have been introduced as substitute of convexity. Here, we consider the multiobjective programming with a set of restrictions indexed in a compact. We ob...

The implementation results of an algorithm designed to solve semi-in&nite programming problems are reported. Due to its computational intensity, parallelisation is used in two stages of the algorithm—for evaluating the global optimum and for checking the feasibility of constraints. The algorithms are parallelised using MPI—the message passing interface. The algorithms are then applied to engine...

A quasi-Newton algorithm for semi-infinite programming using an Leo exact penalty function is described, and numerical results are presented. Comparisons with three Newton algorithms and one other quasi-Newton algorithm show that the algorithm is very promising in practice. AMS classifications: 65K05,90C30.