Semi-infinite Programming Problem Research Articles

Convex SIP Problems with Finitely Representable Compact Index Sets: Immobile Indices and the Properties of the Auxiliary NLP Problem

In the paper, we consider a problem of convex Semi-Infinite Programming with a compact index set defined by a finite number of nonlinear inequalities. While studying this problem, we apply the approach developed in our previous works and based on the notions of immobile indices, the corresponding immobility orders and the properties of a specially constructed auxiliary nonlinear problem. The main results of the paper consist in the formulation of sufficient optimality conditions for a feasible solution of the original SIP problem in terms of the optimality conditions for this solution in a specially constructed auxiliary nonlinear programming problem and in study of certain useful properties of this finite problem.

Nonsmooth semi-infinite minmax programming involving generalized (Φ,ρ)-invexity

This paper introduces some new generalizations of the concept of (Φ,ρ)-invexity for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. These functions are used to derive the necessary and sufficient optimality conditions for a class of nonsmooth semi-infinite minmax programming problems, where set of restrictions are indexed in a compact set. Utilizing the sufficient optimality conditions, the authors formulate three types of dual models and establish weak and strong duality results. The results of the paper extend and unify naturally some earlier results from the literature.

Optimality conditions and duality for nondifferentiable multiobjective semi-infinite programming problems with generalized (C, α, ρ, d)-convexity

This paper obtains sufficient optimality conditions for a nonlinear nondifferentiable multiobjective semi-infinite programming problem involving generalized (C, α, ρ, d)-convex functions. The authors formulate Mond-Weir-type dual model for the nonlinear nondifferentiable multiobjective semi-infinite programming problem and establish weak, strong and strict converse duality theorems relating the primal and the dual problems.

A hybrid method for solving fuzzy semi-infinite linear programming problems

Abilities of fuzzy models in more adaptations with the real world phenomena causes that in this paper, we remodel semi-infinite linear programming problems as fuzzy problems which contained the crisp objective function and the infinite number of fuzzy constraints. Then, we introduce a new hybrid solution method for these fuzzy semi-infinite linear programming problems. This solution technique is a kind of cutting plane algorithm in which its sub-problems were solved by using the Zimmermann-method. Convergence of the presented algorithm is proved, and some numerical test examples are given and also the obtained results are compared.

Parametric Saddle Point Criteria in Semi-Infinite Minimax Fractional Programming Problems Under (p, r)-Invexity

A semi-infinite minimax fractional programming problem with both inequality and equality constraints is considered. Characterizations of an optimal solution by a saddle point of the scalar Lagrange function and the vector-valued Lagrange function defined for such an optimization problem are given. Hence, the equivalence between an optimal solution and a saddle point of the scalar Lagrange function and the vector-valued Lagrange function in the considered semi-infinite minmax fractional programming problem is established under various (p, r)-invexity assumptions.

On strong KKT optimality conditions for multiobjective semi-infinite programming problems with Lipschitzian data

In this paper, we consider a nondifferentiable multiobjective semi-infinite optimization problem. We introduce a qualification condition and derive strong Karusk Kuhn Tucker(KKT) necessary conditions. Then a sufficient optimality condition is proved under invexity assumptions.

A smoothing Levenberg–Marquardt algorithm for semi-infinite programming

In this paper, we present a smoothing Levenberg---Marquardt algorithm for the solution of the semi-infinite programming (SIP) problem. We first reformulate the KKT system of SIP problem into a system of constrained nonsmooth equations. Then we solve this system by a smoothing Levenberg---Marquardt algorithm. The feasibility is ensured via the aggregated constraint, and at each iteration of the presented algorithm only a quadratic programming has to be solved. Global and local superlinear convergence of this algorithm is established under a local error bound condition, which is much weaker than the nonsingularity condition. Preliminary numerical results are reported.

Generalized Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems

This article deals with a generalized semi-infinite programming problem (S). Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a min-max problem constrained with compact convex linked constraints.

Comparative study of RPSALG algorithm for convex semi-infinite programming

The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.

Solving semi-infinite programs by smoothing projected gradient method

In this paper, we study a semi-infinite programming (SIP) problem with a convex set constraint. Using the value function of the lower level problem, we reformulate SIP problem as a nonsmooth optimization problem. Using the theory of nonsmooth Lagrange multiplier rules and Danskin's theorem, we present constraint qualifications and necessary optimality conditions. We propose a new numerical method for solving the problem. The novelty of our numerical method is to use the integral entropy function to approximate the value function and then solve SIP by the smoothing projected gradient method. Moreover we study the relationships between the approximating problems and the original SIP problem. We derive error bounds between the integral entropy function and the value function, and between locally optimal solutions of the smoothing problem and those for the original problem. Using certain second order sufficient conditions, we derive some estimates for locally optimal solutions of problem. Numerical experiments show that the algorithm is efficient for solving SIP.

Duality and Saddle Point for Multiobjective Semi-infinite Programming

This paper is devoted to study the mixed dual models for a class of non-smooth multiobjective semi-infinite programming problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. Weak duality conclusions are derived and proved for mixed type multiobjective dual programs, using the generalized uniform convexity on the functions involved. Some previous duality results for differentiable semi-infinite programming problems turn out to be special cases for the results described in the paper. Furthermore, the vector saddle point theory is discussed for the semi-infinite programming problems. The results extend and improve the corresponding results in the literature.

On a constructive approach to optimality conditions for convex SIP problems with polyhedral index sets

In the paper, we consider a problem of convex Semi-Infinite Programming with an infinite index set in the form of a convex polyhedron. In study of this problem, we apply the approach suggested in our recent paper [Kostyukova OI, Tchemisova TV. Sufficient optimality conditions for convex Semi Infinite Programming. Optim. Methods Softw. 2010;25:279–297], and based on the notions of immobile indices and their immobility orders. The main result of the paper consists in explicit optimality conditions that do not use constraint qualifications and have the form of criterion. The comparison of the new optimality conditions with other known results is provided.

Towards ultrahigh dimensional feature selection for big data

In this paper, we present a new adaptive feature scaling scheme for ultrahigh-dimensional feature selection on Big Data, and then reformulate it as a convex semi-infinite programming (SIP) problem. To address the SIP, we propose an efficient feature generating paradigm. Different from traditional gradient-based approaches that conduct optimization on all input features, the proposed paradigm iteratively activates a group of features, and solves a sequence of multiple kernel learning (MKL) subproblems. To further speed up the training, we propose to solve the MKL subproblems in their primal forms through a modified accelerated proximal gradient approach. Due to such optimization scheme, some efficient cache techniques are also developed. The feature generating paradigm is guaranteed to converge globally under mild conditions, and can achieve lower feature selection bias. Moreover, the proposed method can tackle two challenging tasks in feature selection: 1) group-based feature selection with complex structures, and 2) nonlinear feature selection with explicit feature mappings. Comprehensive experiments on a wide range of synthetic and real-world data sets of tens of million data points with O(1014) features demonstrate the competitive performance of the proposed method over state-of-the-art feature selection methods in terms of generalization performance and training effciency.

A Spline Smoothing Newton Method for Semi-Infinite Minimax Problems

Based on discretization methods for solving semi-infinite programming problems, this paper presents a spline smoothing Newton method for semi-infinite minimax problems. The spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; that is, it introduces an active set technique, so it is more efficient for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems. Numerical tests show that the new method is very efficient.

Constraint Qualifications in Semi-Infinite Systems and Their Applications in Nonsmooth Semi-Infinite Problems with Mixed Constraints

In this paper, we present several constraint qualifications, and we show that conditions guarantee the nonvacuity of the Karush--Kuhn--Tucker multipliers set for nonsmooth semi-infinite programming problems with mixed constraints. The relationships with various constraint qualifications are investigated. All results are given in terms of the Michel--Penot subdifferential.

A new exact penalty method for semi-infinite programming problems

In this paper, we consider a class of nonlinear semi-infinite optimization problems. These problems involve continuous inequality constraints that need to be satisfied at every point in an infinite index set, as well as conventional equality and inequality constraints. By introducing a novel penalty function to penalize constraint violations, we form an approximate optimization problem in which the penalty function is minimized subject to only bound constraints. We then show that this penalty function is exact—that is, when the penalty parameter is sufficiently large, any local solution of the approximate problem can be used to generate a corresponding local solution of the original problem. On this basis, the original problem can be solved as a sequence of approximate nonlinear programming problems. We conclude the paper with some numerical results demonstrating the applicability of our approach to PID control and filter design.

Second-Order Parameter-Free Duality Models in Semi-Infinite Minmax Fractional Programming

In this article, we construct and discuss several second-order parameter-free duality models for a semi-infinite minmax fractional programming problem and establish appropriate duality results under various generalized second-order (ℱ, β, φ, ρ, θ)-univexity assumptions.

Reduction of quantization noise via periodic code for oversampled input signals and the corresponding optimal code design

This paper proposes to reduce the quantization noise using a periodic code, derives a condition for achieving an improvement on the signal to noise ratio (SNR) performance, and proposes an optimal design for the periodic code. To reduce the quantization noise, oversampled input signals are first multiplied by the periodic code and then quantized via a quantizer. The signals are reconstructed via multiplying the quantized signals by the same periodic code and then passing through an ideal lowpass filter. To derive the condition for achieving an improvement on the SNR performance, first the quantization operator is modeled by a deterministic polynomial function. The coefficients in the polynomial function are defined in such a way that the total energy difference between the quantization function and the polynomial function is minimized subject to a specification on the upper bound of the absolute difference. This problem is actually a semi-infinite programming problem and our recently proposed dual parameterization method is employed for finding the globally optimal solution. Second, the condition for improving the SNR performance is derived via a frequency domain formulation. To optimally design the periodic code such that the SNR performance is maximized, a modified gradient descent method that can avoid the obtained solution to be trapped in a locally optimal point and guarantee its convergence is proposed. Computer numerical simulation results show that the proposed system could achieve a significant improvement compared to existing systems such as the conventional system without multiplying to the periodic code, the system with an additive dithering and a first order sigma delta modulator.

Sequential algorithms for structural design optimization under tolerance conditions

A deterministic optimization usually ignores the effects of uncertainties in design variables or design parameters on the constraints. In practical applications, it is required that the optimum solution can endure some tolerance so that the constraints are still satisfied when the solution undergoes variations within the tolerance range. An optimization problem under tolerance conditions is formulated in this article. It is a kind of robust design and a special case of a generalized semi-infinite programming (GSIP) problem. To overcome the deficiency of directly solving the double loop optimization, two sequential algorithms are then proposed for obtaining the solution, i.e. the double loop optimization is solved by a sequence of cycles. In each cycle a deterministic optimization and a worst case analysis are performed in succession. In sequential algorithm 1 (SA1), a shifting factor is introduced to adjust the feasible region in the next cycle, while in sequential algorithm 2 (SA2), the shifting factor is replaced by a shifting vector. Several examples are presented to demonstrate the efficiency of the proposed methods. An optimal design result based on the presented method can endure certain variation of design variables without violating the constraints. For GSIP, it is shown that SA1 can obtain a solution with equivalent accuracy and efficiency to a local reduction method (LRM). Nevertheless, the LRM is not applicable to the tolerance design problem studied in this article.

Optimality conditions for nonsmooth semi-infinite multiobjective programming

This paper is devoted to the study of nonsmooth multiobjective semi-infinite programming problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. We introduce several kinds of constraint qualifications for these problems, and then necessary optimality conditions for weakly efficient solutions are investigated. Finally by imposing assumptions of generalized convexity we give sufficient conditions for efficient solutions.