Explicit Optimality Conditions Research Articles

Optimality Conditions in Control Problems with Random State Constraints in Probabilistic or Almost Sure Form

In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. Although the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them with a model based on robust constraints with respect to the (compact) support of the given distribution. Funding: The authors thank the Deutsche Forschungsgemeinschaft [Projects B02 and B04 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”] for support. C. Geiersbach acknowledges support from the Deutsche Forschungsgemeinschaft [Germany’s Excellence Strategy–the Berlin Mathematics Research Center MATH+ Grant EXC-2046/1, Project 390685689]. R. Henrion acknowledges support from the Fondation Mathématique Jacques Hadamard [Program Gaspard Monge in Optimization and Operations Research, including support to this program by Electricité de France].

The continuous delayed distribution problem

Nowadays, decision makers (DMs) at companies have access to extensive and accurate data, which means they have the opportunity to grow and improve if they use the latent potential effectively. We address the complex problem of optimizing decisions in a multi-period one-warehouse multi-retailer inventory system with stochastic continuous demand and an option of delayed distributions. At the beginning of each period, the DM determines each retailer’s target stock levels, as well as the number of items to be held back at a central location for later distribution(s). Such a policy offers partial inventory pooling through the holdback quantity. The decisions in each period are data-driven, i.e., made based on sales data available through an information system up to that point in time.We model the problem as a multi-stage stochastic program with recourse. For the general case, we develop a new recursive solution algorithm, which is based on subgradient optimization and an analysis of system dynamics. For the special case of two identical retailers and two periods, we provide explicit optimality conditions based on the subgradients. Using a large numerical study, we evaluate the performance of our proposed policy and compare it to two benchmark policies. We also demonstrate the impact of various problem parameters on the optimal solution and objective value.

Convex duality in nonlinear optimal transport

This article studies problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of a large class of related problems in probability theory and allows for generalizations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems. When the objective takes the form of a convex integral functional, we obtain more explicit optimality conditions and establish the existence of solutions for a relaxed formulation of the problem. This covers, in particular, the mass transportation problem and its nonlinear generalizations.

Optimality conditions for linear copositive programming problems with isolated immobile indices

ABSTRACTIn the present paper, we apply our recent results on optimality for convex semi-infinite programming to a problem of linear copositive programming (LCP). We prove explicit optimality conditions that use concepts of immobile indices and their immobility orders and do not require the Slater constraint qualification to be satisfied. The only assumption that we impose here is that the set of immobile indices consists of isolated points and hence is finite. This assumption is weaker than the Slater condition; therefore, the optimality conditions obtained in the paper are more general when compared with those usually used in LCP. We present an example of a problem in which the new optimality conditions allow one to test the optimality of a given feasible solution while the known optimality conditions fail to do this. Further, we use the immobile indices to construct a pair of regularized dual copositive problems and show that regardless of whether the Slater condition is satisfied or not, the duality gap between the optimal values of these problems is zero. An example of a problem is presented for which the standard strict duality fails, but the duality gap obtained by using the regularized dual problem vanishes.

Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets

In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems.

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.

Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions

We present explicit optimality conditions for a nonsmooth functional defined over the (properly or weakly) Pareto set associated with a multi-objective linear-quadratic control problem. This problem is very difficult even in a finite dimensional setting , i.e. when, instead of a control problem, we deal with a mathematical programming problem. Amongst various applications, our problem may be considered as a response for a decision maker when he has to choose a solution over the solution set of the grand coalition $p$-player cooperative differential game.

Study of a special nonlinear problem arising in convex semi-infinite programming

We consider convex problems of semi-infinite programming (SIP) using an approach based on the implicit optimality criterion. This criterion allows one to replace optimality conditions for a feasible solution x 0 of the convex SIP problem by such conditions for x 0 in some nonlinear programming (NLP) problem denoted by NLP(I(x 0)). This nonlinear problem, constructed on the base of special characteristics of the original SIP problem, so-called immobile indices and their immobility orders, has a special structure and a diversity of important properties. We study these properties and use them to obtain efficient explicit optimality conditions for the problem NLP(I(x 0)). Application of these conditions, together with the implicit optimality criterion, gives new efficient optimality conditions for convex SIP problems. Special attention is paid to SIP problems whose constraints do not satisfy the Slater condition and to problems with analytic constraint functions for which we obtain optimality conditions in the form of a criterion. Comparison with some known optimality conditions for convex SIP is provided.

An inventory system with periodic regular review and flexible emergency review

We analyze a discrete time inventory model with two supply modes. Regular orders are placed periodically and arrive after a deterministic lead time. Emergency orders, characterized by a shorter deterministic lead time but higher purchase costs, can be placed in case of imminent stock-outs. We propose and analyze flexible order-up-to policies with emergency orders that arrive one, two, ⃛, up to some fixed number of time units before a regular order arrives. Based on an approximate cost model, we develop two sets of approximate explicit optimality conditions. The first set of conditions are rather complicated, but simulation shows that they do lead to near-optimal policies in most cases. The second set of conditions are very simple and seem more practical, but they only lead to a near-optimal policy if there are enough emergency supply opportunities in a review period to prevent most backorders.

An inventory system with periodic regular review and flexible emergency review

We analyze a discrete time inventory model with two supply modes. Regular orders are placed periodically and arrive after a deterministic lead time. Emergency orders, characterized by a shorter deterministic lead time but higher purchase costs, can be placed in case of imminent stock-outs. We propose and analyze flexible order-up-to policies with emergency orders that arrive one, two, ..., up to some fixed number of time units before a regular order arrives. Based on an approximate cost model, we develop two sets of approximate explicit optimality conditions. The first set of conditions are rather complicated, but simulation shows that they do lead to near-optimal policies in most cases. The second set of conditions are very simple and seem more practical, but they only lead to a near-optimal policy if there are enough emergency supply opportunities in a review period to prevent most backorders.

Shape optimization in laminated composite plates

At this stage a substantial body of literature exists for shape optimization involving isotropic materials. The reader is referred to the articles of Haug [1] and Haftka and Grandhi [2] for overviews of the material. One approach to the problem is analytical. Calculus of variation techniques are employed to derive explicit optimality conditions. Choi and Haug [3], Dems and Mroz [4], and Banichuk [5, 6] have adopted this approach. Of note is the recent work of Kikuchi, Chung, Torigaki and Taylor [7] who used variational methods together with an adaptive grid scheme developed by Diaz, Kikuchi and Taylor [8] for shape optimization in linear elastic structures. The works of Braibant and Fleury [9, 10], involving B-splines, should also be cited. Up to now, structural optimization of laminated composites has involved ply orientations and thicknesses as design variables. To the authors' knowledge, no work on shape optimization Of laminated structures has been done. Kicher and Chao [11] considered the optimum design of fiber composite cylinders under combined axial, radial, and torsional loads. Minimum weight was taken as the objective function, with the ply thickness being the design variable. Hirano [12, 13], Park [14], and Tauchert and Adibhatla [15] treated the optimum design of laminated plates under various types of loading. The objective functions used were maximum buckling load, first ply failure (see [16]), and minimum strain energy, respectively. Hirano and Park used fiber angles as design variables, whereas Tauchert and Adibhatla employed both fiber angles and layer thickness as variables. Yiping [17] treated the optimal design of a laminated plate with several elliptical holes under uniaxial tensile loading. Minimizing the maximum value of the Tsai-Wu failure functional (see [18]) was the objective and the fiber angle was the design variable. In all of the above works on composites, mathematical programming techniques were used to obtain the solutions. Other approaches have been used. Bauchau [19] considered the optimal design of rotating graphite-epoxy shafts.

An iterative approach to strucural optimization

AbstractAn iterative procedure of structural optimization is discussed that combines finite element analysis with the use of explicit optimality conditions. The procedure is illustrated by examples concerning sandwich beams with constraints on the deflection at a specified section, the maximum deflection or the sum of the maximum deflection in all spans of a continuous beam. The examples suggest that, starting from a uniform beam, by far the greatest saving in weight is obtained in the first two or three iterations.

Minimum-weight design of continuous beams under displacement and stress constraints

Explicit optimality conditions for minimum-weight design of elastic sandwich beams with segmentwise constant structural stiffness, subject to displacement and mean-square stress constraints, are obtained. An iterative procedure that combines the use of the optimality conditions with finite-element analysis is proposed and is illustrated by numerical examples. These examples suggest that very few iterations are necessary to obtain a good approximation to the optimal design. It is shown that, for practical purposes, the optimization problem may be simplified by using the optimality conditions derived for statically determinate beams instead of those valid for statically indeterminate beams.