Semi-infinite Programming Research Articles

A new approach for worst-case regret portfolio optimization problem

This paper considers the worst-case regret portfolio optimization problem when the distributions of the asset returns are uncertain. In general, the solution to this problem is NP hard and approximation methods that minimise the difference between the maximum return and the sum of each portfolio return are often proposed. Applying the duality of semi-infinite programming, the worst-case regret portfolio optimization problem with uncertain distributions can be equivalently reformulated to a linear optimization problem, and the established solution approaches for linear optimization can then be applied. An example of a portfolio optimization problem is provided to show the efficiency of our method and the results demonstrate that our method can satisfy the portfolio risk diversification property under the uncertain distributions of the returns.

Problem-based optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem.

Strong Karush–Kuhn–Tucker optimality conditions for multiobjective semi-infinite programming via tangential subdifferential

The main aim of this paper is to study strong Karush–Kuhn–Tucker (KKT) optimality conditions for nonsmooth multiobjective semi-infinite programming (MSIP) problems. By using tangential subdifferential and suitable regularity conditions, we establish some strong necessary optimality conditions for some types of efficient solutions of nonsmooth MSIP problems. In addition to the theoretical results, some examples are provided to illustrate the advantages of our outcomes.

Method of Multiobjective Optimization of Controlled Systems with Distributed Parameters

The constructive method for multicriteria optimization of control processes of deterministic and not fully defined controlled systems with distributed parameters, described by linear multidimensional parabolic partial differential equations with internal and boundary control actions, is proposed. The optimization problem is considered when an uniform approximation accuracy of object’s final state to the required spatial distribution of controlled function is given. The suggested approach is based on the one-criterion option in the form of minimax convolution of the normalized quality criteria and the subsequent transition to the equivalent form of a typical variational problem with constraints. It is applied to the deterministic model of an object described by an infinite system of differential equations with respect to time-dependent modes of the controlled quantity expansion in a series of eigenfunctions of the initial-boundary value problem. Further procedures for the preliminary parametrization of control actions, based on analytical optimum conditions and reduction to semi-infinite programming problems, allow one to find the desired extremals using their Chebyshev properties and fundamental laws of the domain in typical application conditions of estimating the accuracy of approaching the object’s final state to the required one in a uniform metric. The obtained results are extended to the tasks of program control on the principle of guaranteed result by ensembles of object trajectories under conditions of interval uncertainty of the parametric characteristics of the distributed system and multiple external disturbances. A demonstrated example of a multicriteria optimization of an innovation technology of metal induction heating prior a hot forming is of special interest. The typical optimization criteria such as energy consumption, metal loss due to scale formation, and heating accuracy are considered as components of vector optimization criterion.

Duality models for multiobjective semiinfinite fractional programming problems involving type-I and related functions

In this paper, we consider several dual models for a multiobjective semiinfinite fractional programming problem and establish usual duality theorems involving type-I and related functions. Our results generalize the results of Zalmai and Zhang [G.J. Zalmai and Qing-hong Zhang, Parametric duality models for semiinfinite multiobjective fractional programming problems containing generalized (α, η, ρ)-V-invex functions, Acta Math. Appl. Sinica 29(2) (2013), 225–240]. Results obtained in this paper extend and unify several results in the literature.

Set-membership nonlinear regression approach to parameter estimation

This paper introduces set-membership nonlinear regression (SMR), a new approach to nonlinear regression under uncertainty. The problem is to determine the subregion in parameter space enclosing all (global) solutions to a nonlinear regression problem in the presence of bounded uncertainty on the observed variables. Our focus is on nonlinear algebraic models. We investigate the connections of SMR with (i) the classical statistical inference methods, and (ii) the usual set-membership estimation approach where the model predictions are constrained within bounded measurement errors. We also develop a computational framework to describe tight enclosures of the SMR regions using semi-infinite programming and complete-search methods, in the form of likelihood contour and polyhedral enclosures. The case study of a parameter estimation problem in microbial growth is presented to illustrate various theoretical and computational aspects of the SMR approach.

Isolated efficiency in nonsmooth semi-infinite multi-objective programming

ABSTRACTIn this paper, isolated efficient solutions of a given nonsmooth Multi-Objective Semi-Infinite Programming problem (MOSIP) are studied. Two new Data Qualifications (DQs) are introduced and it is shown that these DQs are, to a large extent, weaker than already known Constraint Qualifications (CQs). The relationships between isolated efficiency and some relevant notions existing in the literature, including robustness, are established. Various necessary and sufficient conditions for characterizing isolated efficient solutions of a general problem are derived. It is done invoking the tangent cones, the normal cones, the generalized directional derivatives, and some gap functions. Using these characterizations, the (strongly) perturbed Karush-Kuhn-Tucker (KKT) optimality conditions for MOSIP are analyzed. Furthermore, it is shown that each isolated efficient solution is a Geoffrion properly efficient solution under appropriate assumptions. Moreover, Kuhn-Tucker (KT) and Klinger properly efficient solutions for a nonsmooth MOSIP are defined and it is proved that each isolated efficient solution is a KT properly efficient solution in general, and a Klinger properly efficient solution under a DQ. Finally, in the last section, the largest isolated efficiency constant for a given isolated efficient solution is determined.

Characterization of Isolated Efficient Solutions in Nonsmooth Multiobjective Semi-infinite Programming

The purpose of this paper is to characterize the isolated efficient solutions of a given nonsmooth multiobjective optimization problem with finitely many objective functions and infinitely many constraints. To do this, we introduce a new class of functions in order to get Karush–Kuhn–Tucker type necessary and sufficient optimality conditions for the problem. This article can be considered as a supplement to the recent paper by Kanzi (Iran J Sci Technol Trans Sci 42:1537–1544, 2018).

Robust power allocation for two-tier heterogeneous networks under channel uncertainties

In this paper, the trade-off among system sum energy consumption and robustness is studied. In this regard, a robust power allocation problem is formulated for a two-tier heterogeneous network with uplink transmission mode and consideration of imperfect channel state information. The objective is to minimize the total transmit power of femtocell users (FUs), while the interference to macrocell user receiver is limited to a predefined interference level, the transmit power of each FU transmitter is kept within their power budgets, and the actual signal-to-interference-plus-noise ratio of each femtocell receiver is above a minimum threshold. Considering the uncertainties of the interference links from FUs to macrocell base stations and forward transmission links of each FU, the robust power allocation problem is formulated as a semi-infinite programming problem (SIPP). By the worst-case approach, the SIPP is transformed into a convex optimization problem solved by the Lagrange dual decomposition method. Moreover, the feasible regions of constraints, computational complexity, and sensitivity degree of the proposed robust algorithm are also analyzed. Simulation results investigate the impact of channel uncertainties and the superiority of the proposed algorithm by comparing with non-robust algorithm.

Global optimization of generalized semi-infinite programs using disjunctive programming

We propose a new branch-and-bound algorithm for global minimization of box-constrained generalized semi-infinite programs. It treats the inherent disjunctive structure of these problems by tailored lower bounding procedures. Three different possibilities are examined. The first one relies on standard lower bounding procedures from conjunctive global optimization as described in Kirst et al. (J Global Optim 69: 283–307, 2017). The second and the third alternative are based on linearization techniques by which we derive linear disjunctive relaxations of the considered sub-problems. Solving these by either mixed-integer linear reformulations or, alternatively, by disjunctive linear programming techniques yields two additional possibilities. Our numerical results on standard test problems with these three lower bounding procedures show the merits of our approach.

On First-Order Conditions for Optimality of Nondifferentiable Semi-infinite Programming

The purpose of this paper is to give some new Karush–Kuhn–Tucker-type necessary optimality conditions for nonsmooth semi-infinite problems. Moreover, we present some suitable examples for our results. The paper is organized by Frechet and Mordukhovich subdifferentials.

Filter Trust Region Method for Nonlinear Semi-Infinite Programming Problem

We present a filter trust region method for nonlinear semi-infinite programming. Based on the discretization technique and motivated by the multiobjective programming, we transform the semi-infinite problem into a finite one. Together with the filter technique, we propose a modified method that avoids the merit function. Compared with the existing methods, our method is more flexible and easier to implement. Under some mild conditions, the convergent properties are proved. Moreover, the numerical results are reported in the end.

Solvability And Primal-dual Partitions Of The Space Of Continuous Linear Semi-infinite Optimization Problems

Different partitions of the parameter spaceof all linear semi-infinite programming problems witha fixed compact set of indices and continuous rightand left hand side coefficients have been consideredin this paper. The optimization problems are classifiedin a different manner, e.g., consistent and inconsistent, solvable (with bounded optimal value and nonempty optimal set), unsolvable (with bounded optimal valueand empty optimal set) and unbounded (with infinite optimal value). The classification we propose generatesa partition of the parameter space, called second general primal-dual partition. We characterize each cell of the partition by means of necessary and sufficient, and in some cases only necessary or sufficient conditions, assuring that the pair of problems (primal and dual) belongs to that cell. In addition, we show non emptiness of each cell of the partition and with plenty of examples we demonstrate that some of the conditions are only necessary or sufficient. Finally, we investigate various questions of stability of the presented partition.

Robust dynamic optimization of batch processes under parametric uncertainty: Utilizing approaches from semi-infinite programs

The optimal solution in dynamic optimization of batch processes often exhibits active path constraints. The goal of this work is the robust satisfaction of path constraints in the presence of parametric uncertainties based on known worst-case formulations. These formulations are interpreted as semi-infinite programs (SIP). Two known SIP algorithms are extended to the dynamic case and assessed. One is a discretization approach and the other a local reduction approach. With these presented concepts, robust path constraint satisfaction is in principle guaranteed. In this work, however, local methods are used to approximate the global solution of the lower-level problem with local solvers thus allowing for (rather unlikely) constraint violations. Finally, the penicillin fermentation is introduced as a well-known case study with uncertainties, which is modified in this work by adding further dependencies. The adaptation of the SIP concepts to dynamic optimization problems are shown to be successful for this case study.

A dynamic game approach to distributionally robust safety specifications for stochastic systems

This paper presents a new safety specification method that is robust against errors in the probability distribution of disturbances. Our proposed distributionally robust safe policy maximizes the probability of a system remaining in a desired set for all times, subject to the worst possible disturbance distribution in an ambiguity set. We propose a dynamic game formulation of constructing such policies and identify conditions under which a non-randomized Markov policy is optimal. Based on this existence result, we develop a practical design approach to safety-oriented stochastic controllers with limited information about disturbance distributions. However, an associated Bellman equation involves infinite-dimensional minimax optimization problems since the disturbance distribution may have a continuous density. To alleviate computational issues, we propose a duality-based reformulation method that converts the infinite-dimensional minimax problem into a semi-infinite program that can be solved using existing convergent algorithms. We prove that there is no duality gap, and that this approach thus preserves optimality. The results of numerical tests confirm that the proposed method is robust against distributional errors in disturbances, while a standard stochastic safety verification tool is not.

Optimality, scalarization and duality in linear vector semi-infinite programming

In this paper, we introduce some qualification conditions for a linear vector semi-infinite programming problem. The new qualification conditions are used for development of necessary conditions for weak efficient, isolated efficient and -efficient solutions of such a problem. Sufficient conditions for the existence of such solutions are also provided. In addition, a new general scalarization technique for solving the reference problem is presented. Finally, we propose a type of Wolf dual problem and examine weakstrong duality relations.

Optimality conditions for semi-infinite programming problems involving generalized convexity

We apply some advanced tools of quasiconvex analysis to establish Karush–Kuhn–Tucker type necessary and sufficient optimality conditions for non-differentiable semi-infinite programming problems. In addition, we propose a linear characterization of optimality for the mentioned problems. Examples are also designed to analyze and illustrate the results obtained.

An infeasible bundle method for nonconvex constrained optimization with application to semi-infinite programming problems

The main difficulty for solving semi-infinite programming (SIP) problem is precisely that it has infinitely many constraints. By using a maximum function, the SIP problem can be rewritten as a nonconvex nonsmooth constrained optimization (NNCO) problem. Global convergence in most of constrained optimization algorithms has traditionally been enforced by the use of a penalty function or filter strategy. In this paper, we propose an infeasible bundle method for NNCO problem based on the so-called improvement functions, without a penalty function and filter strategy. The method appears to be more direct and easier to implement, in the sense that it is closer in spirit and structure to the well-developed unconstrained bundle methods. Under a special constraint qualification, the sequence generated by this algorithm converges to the KKT point of the NNCO problem as well as the SIP problems. Preliminary numerical results show that this algorithm is robust and efficient for NNCO problems and SIP problems.

Bounded-error optimal experimental design via global solution of constrained min–max program

We present an improvement of existing methods for globally solving optimal experimental design (OED) for bounded-error estimation based on a bilevel formulation from Mukkala et al. (2017). The proposed solution method for the min–max program is based on our method for generalized semi-infinite programs (via restriction of the right-hand side). The algorithm employed has the advantage that it guarantees a global solution for the OED assuming the global solution of two subproblems. To obtain a feasible solution only the lower-level problem has to be solved globally. In case of a local solution of the upper-level problem, the solution is still feasible though it is an upper bound of the global solution. The min–max method for OED is illustrated with four examples: two simple chemical reactions, BET-adsorption and a reformulated predator-prey system. The benefits of global methods are shown along with the limitations of state-of-the-art global solvers.

A Multi-objective Method for Solving Fuzzy Linear Programming Based on Semi-infinite Model

ABSTRACTIn this paper, we present a new method to solve a fuzzy linear programming problem with fuzzy coefficients in the constraints and the objective function based on solving an associated multi-objective model. In particular, we present a weighted method for linear semi-infinite programming (LSIP) model to solve the original problem. Finally, a numerical example is included to illustrate the suggested solving process.