Semi-infinite Programming Research Articles

Primal-dual path following method for nonlinear semi-infinite programs with semi-definite constraints

In this paper, we propose a primal-dual path following method for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short. A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs). However, it is often too demanding to find exact solutions of SDPs. In contrast, our approach does not rely on solving SDPs accurately but approximately following a path leading to a solution, which is formed on the intersection of the semi-infinite feasible region and the interior of the semi-definite feasible region. Specifically, we first present a prototype path-following method and show its global weak* convergence to a Karush-Kuhn-Tucker point of the SISDP under some mild assumptions. Next, to achieve fast local convergence, we integrate a two-step sequential quadratic programming method equipped with the Monteiro-Zhang scaling technique into the prototype method. We prove two-step superlinear convergence of the resulting algorithm using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs obtained by finite relaxation of the SISDP.

Optimal Power Flow in DC Networks With Robust Feasibility and Stability Guarantees

With high penetrations of renewable generation and variable loads, there is significant uncertainty associated with power flows in dc networks such that stability and operational constraint satisfaction are of concern. Most existing dc network optimal power flow (DN-OPF) formulations assume exact knowledge of loading conditions and do not provide stability guarantees. In contrast, this article studies a DN-OPF formulation, which considers both stability and operational constraint satisfaction under uncertainty. The need to account for a range of uncertainty realizations in this article’s robust optimization formulation results in a challenging semi-infinite program (SIP). The proposed solution algorithm reformulates this SIP into a computationally tractable problem by constructing a tight convex inner approximation of the feasible region using sufficient conditions for the existence of a feasible and stable power flow solution. Optimal generator setpoints are obtained by optimizing over the proposed convex stability set. The validity and value of the proposed algorithm are demonstrated through various dc networks adapted from IEEE test cases.

Conic Linear Programming Duals for Classes of Quadratic Semi-Infinite Programs with Applications

Conic Linear Programming Duals for Classes of Quadratic Semi-Infinite Programs with Applications

Dual Set-Membership State Estimation for Power Distribution Networks Under Event-Triggered Mechanism

This article is concerned with the set-membership state estimation problem for power distribution networks (PDNs) over a resource-constrained communication network under the influence of unknown but bounded (UBB) noises. Firstly, in order to alleviate the pressure of information communication network (ICN) while meeting the state estimation requirements, the event-triggered mechanism is adopted to send data containing more valid information to estimation center, reasonably reducing the signal transmission frequency. Secondly, an event-triggered dual set-membership filter (ET-DSMF) is designed to improve the performance of state estimation. The proposed filter performs a discrete approximation for a semi-infinite programming problem by the random sampling technique, and a compact linearization error bounding ellipsoid is obtained by solving the dual problem of the nonlinear programming. Subsequently, a sufficient condition for the existence of the estimated ellipsoid is derived depending on the mathematical induction method. The key time-varying filter gain matrix and optimal estimated ellipsoid are determined recursively by solving a convex optimization problem, according to the minimum trace criterion. Finally, the effectiveness of the proposed filtering algorithm is demonstrated by performing simulation experiments on the IEEE 13 distribution test system.

Solving continuous set covering problems by means of semi-infinite optimization

This article introduces the new class of continuous set covering problems. These optimization problems result, among others, from product portfolio design tasks with products depending continuously on design parameters and the requirement that the product portfolio satisfies customer specifications that are provided as a compact set. We show that the problem can be formulated as semi-infinite optimization problem (SIP). Yet, the inherent non-smoothness of the semi-infinite constraint function hinders the straightforward application of standard methods from semi-infinite programming. We suggest an algorithm combining adaptive discretization of the infinite index set and replacement of the non-smooth constraint function by a two-parametric smoothing function. Under few requirements, the algorithm converges and the distance of a current iterate can be bounded in terms of the discretization and smoothing error. By means of a numerical example from product portfolio optimization, we demonstrate that the proposed algorithm only needs relatively few discretization points and thus keeps the problem dimensions small.

Duality for convex infinite optimization on linear spaces

AbstractThis note establishes a limiting formula for the conic Lagrangian dual of a convex infinite optimization problem, correcting the classical version of Karney [Math. Programming 27 (1983) 75-82] for convex semi-infinite programs. A reformulation of the convex infinite optimization problem with a single constraint leads to a limiting formula for the corresponding Lagrangian dual, called sup-dual, and also for the primal problem in the case when strong Slater condition holds, which also entails strong sup-duality.

Characterizing robust optimal solution sets for nonconvex uncertain semi-infinite programming problems involving tangential subdifferentials

Characterizing robust optimal solution sets for nonconvex uncertain semi-infinite programming problems involving tangential subdifferentials

Robust optimality conditions and duality for nonsmooth multiobjective fractional semi-infinite programming problems with uncertain data

In this article, some Karush-Kuhn-Tucker type robust optimality conditions and duality for an uncertain nonsmooth multiobjective fractional semi-infinite programming problem ((UMFP), for short) are established. First, we provide, by combining robust optimization and the robust limiting constraint qualification, robust necessary optimality conditions in terms of Mordukhovich's subdifferentials. Under suitable assumptions on the generalized convexity/the strictly generalized convexity, robust necessary optimality condition becomes robust sufficient optimality condition. Second, we formulate types of Mond-Weir and Wolfe robust dual problem for (UMFP) via the Mordukhovich subdifferentials. Finally, as an application, we establish weak/strong/converse robust duality theorems for the problem (UMFP) and its Mond-Weir and Wolfe types dual problem. Some illustrative examples are also provided for our findings.

On approximate quasi Pareto solutions in nonsmooth semi-infinite interval-valued vector optimization problems

This paper deals with approximate solutions of a nonsmooth semi-infinite programming with multiple interval-valued objective functions. We first introduce four types of approximate quasi Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these approximate solutions. Sufficient conditions for approximate quasi Pareto solutions of such a problem are also provided by means of introducing the concepts of approximate (strictly) pseudo-quasi generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, a Mond–Weir type dual model in approximate form is formulated, and weak, strong and converse-like duality relations are proposed.

A robust multi-supplier multi-period inventory model with uncertain market demand and carbon emission constraint

The multi-period inventory management determines the optimal order quantity of products at the beginning of each period, which is an important research issue in supply chain management. In this paper, we propose a robust multi-supplier multi-period inventory model with uncertain market demand and uncertain unit carbon emission in transportation. We select product quality, ordering cost, service level, and emergency capacity as evaluation criteria for each supplier, employ analytic hierarchy process (AHP) technique to comprehensively evaluate and score each supplier. According to the evaluated score, the order weight for each supplier is obtained. In our model, we develop carbon emission constraint based on the carbon cap and carbon trading mechanism. As for uncertain market demand and uncertain unit carbon emission in transportation, we construct Box+Ball and Budget uncertainty sets to characterize various model uncertainties. The robust inventory model is a semi-infinite programming model, and cannot be solved directly. We transform the semi-infinite programming model into a mixed-integer second-order cone-programming (MISOCP) one via cone duality theory. Finally, the effectiveness of the proposed optimization method is illustrated by a case study.

Necessary and Sufficient Optimality Conditions for Semi-infinite Programming with Multiple Fuzzy-valued Objective Functions

This paper deals with semi-infifinite programming with multiple fuzzy-valued objective functions. Firstly, some types of effificient solutions are proposed and illustrated in some examples. Then, necessary and suffificient Karush-Kuhn-Tucker optimality conditions for semi-infifinite programming with multiple fuzzy-valued objective functions are established.

Karush-Kuhn-Tucker optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints

Karush-Kuhn-Tucker optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints

Corrigendum on: Optimality conditions and duality for multiobjective semi-infinite programming with data uncertainty via Mordukhovich subdifferential (Yugoslav Journal of Operations Research, vol. 31, no 4, doi: https://doi.org/10.2298/YJOR201017013P)

The author of the article "Optimality Conditions and Duality for Multiobjective Semi-Infinite Programming with Data Uncertainity via Mordukhovich Subdifferential", Thanh-Hung Pham has informed the Editor about necessary corrections of the paper. <br><br><font color="red"><b> Link to the corrected article <u><a href="http://dx.doi.org/10.2298/YJOR201017013P">10.2298/YJOR201017013P</a></b></u>

Approximate optimality conditions and approximate duality theorems for nonlinear semi-infinite programming problems with uncertainty data

Approximate optimality conditions and approximate duality theorems for nonlinear semi-infinite programming problems with uncertainty data

Robust mechanism design and production structure for assembly systems with asymmetric cost information

We consider the design of a robust procurement mechanism for an assembler who must procure multiple complementary components under limited information regarding the unit production costs of the potential suppliers. We abandon the standard assumption that there exists a common prior distribution for the suppliers’ costs and we employ the max-min criterion in which we design a mechanism to maximize the assembler’s worst-case expected profit when only the means of the suppliers’ costs are known. We formulate this problem using a linear semi-infinite programming model and we apply a primal-dual approach to reduce the problem to a single joint optimization model. We characterize the optimal procurement mechanism (payments and ordering quantities) under the assumption of balanced ordering and uncertain end-customer demand. We use these results to compare the performance of component production, under which the components are sourced from separate suppliers, and integrated production, under which the components are sourced from the same supplier. We find that component production is preferred when the production costs of the components are heterogeneous. We also propose two heuristic mechanisms which are easier to compute and implement in practice, particularly when the number of components is large. Finally, we extend our analysis to the case in which the average and variance of the suppliers’ costs are known.

Necessary optimality conditions for a multiobjective semi-infinite interval-valued programming problem

Necessary optimality conditions for a multiobjective semi-infinite interval-valued programming problem

On Constrained Set-Valued Semi-Infinite Programming Problems with ρ-Cone Arcwise Connectedness

In this paper, we establish sufficient Karush–Kuhn–Tucker (for short, KKT) conditions of a set-valued semi-infinite programming problem (SP) via the notion of contingent epiderivative of set-valued maps. We also derive duality results of Mond–Weir (MWD), Wolfe (WD), and mixed (MD) types of the problem (SP) under ρ-cone arcwise connectedness assumptions.

E-optimum sensor selection for estimation of subsets of parameters

The design of a network of observation locations is addressed in the setting of estimating unknown parameters of a spatiotemporal system modelled by a partial differential equation. Interest is in estimating only a subset of these parameters as accurately as possible. The other parameters, called nuisance parameters, must also be estimated although we are interested in neither their values, nor accuracies. The maximal eigenvalue of the covariance matrix of the maximum-likelihood estimator of the parameters of interest is used as the measure of the identification accuracy. In order to make selection of a best subset of gauged sites from a possibly very large set of candidate sites computationally tractable, its convex relaxation is introduced. Two major problems to be tackled are the potential singularity of the optimal information matrix associated with all unknown parameters and the nondifferentiability of the optimality criterion. The former is settled by imposing a constraint on the minimal allowable value of the determinant of the information matrix. The latter is resolved by reformulating the problem as a convex semi-infinite programming problem whose solution is sought by solving a sequence of finite low-dimensional min–max problems using extremely efficient generalized simplicial decomposition. The excellent performance of the proposed technique is illustrated by an example involving optimal sensor node activation in a large sensor network collecting measurements to identify a moving contaminating source.

Optimality Conditions and Duality for Multiobjective Semi-infinite Programming on Hadamard Manifolds

Optimality Conditions and Duality for Multiobjective Semi-infinite Programming on Hadamard Manifolds

On Minimax Fractional Semi-Infinite Programming Problems with Applications

Using some advanced tools of variational analysis and generalized differentiation, we establish necessary conditions for locally optimal solutions of minimax fractional programming problems under the limiting constraint qualification (LCQ). Sufficient conditions for such solutions to the considered problem are also provided by introducing generalized convex functions defined in terms of the limiting subdifferential for locally Lipschitz functions. In addition, some duality results for minimax fractional programming problems are also provided. Finally, by using the obtained results, we derive necessary and sufficient conditions for weak Pareto solutions to the multiobjective semi-infinite fractional optimization problem.