Uniform saturation in linear inequality systems

Redundant constraints in linear inequality systems can be characterized as those inequalities that can be removed from an arbitrary linear optimization problem posed on its solution set without modifying its value and its optimal set. A constraint is saturated in a given linear optimization problem when it is binding at the optimal set. Saturation is a property related with the preservation of the value and the optimal set under the elimination of the given constraint, phenomena which can be seen as weaker forms of excess information in linear optimization problems. We say that an inequality of a given linear inequality system is uniformly saturated when it is saturated for any solvable linear optimization problem posed on its solution set. This paper characterizes the uniform saturated inequalities and other related classes of inequalities. This work was supported by the MCYT of Spain and FEDER of UE, Grant BFM2002-04114-C02-01.     

On the optimal value function for certain linear programs with unbounded optimal solution sets

Often, the coefficients of a linear programming problem represent estimates of true values of data or are subject to systematic variations. In such cases, it is useful to perturb the original data and to either compute, estimate, or otherwise describe the values of the functionf which gives the optimal value of the linear program for each perturbation. If the right-hand derivative off at a chosen point exists and is calculated, then the values off in a neighborhood of that point can be estimated. However, if the optimal solution set of either the primal problem or the dual problem is unbounded, then this derivative may not exist. In this note, we show that, frequently, even if the primal problem or the dual problem has an unbounded optimal solution set, the nature of the values off at points near a given point can be investigated. To illustrate the potential utility of our results, their application to two types of problems is also explained.This research was supported, in part, by the Center for Econometrics and Decision Sciences, University of Florida, Gainesville, Florida.The author would like to thank two anonymous reviewers for their most useful comments on earlier versions of this paper.     

Dismantle or remanufacture?

In this paper we study a firm’s disposition decision for returned end-of-use products, which can either be remanufactured and sold, or dismantled into parts that can be reused. We formulate this problem as a multi-period stochastic dynamic program, and find the structure of the optimal policy, which consists of monotonic switching curves. Specifically, if it is optimal to remanufacture in a given period and for given inventory levels, then it is also optimal to remanufacture when the inventory of part(s) is higher or the inventory of remanufactured product is lower.     

Directional differentiability of the optimal value function in convex semi-infinite programming

In this paper, directional differentiability properties of the optimal value function of a parameterized semi-infinite programming problem are studied. It is shown that if the unperturbed semi-infinite programming problem is convex, then the corresponding optimal value function is directionally differentiable under mild regularity assumptions. A max-min formula for the directional derivatives, well-known in the finite convex case, is given.     

A superharmonic approach to solving infinite horizon partially observable Markov decision problems

In this paper we use an approach which uses a superharmonic property of a sequence of functions generated by an algorithm to show that these functions converge in a non-increasing manner to the optimal value function for our problem, and bounds are given for the loss of optimality if the computational process is terminated at any iteration. The basic procedure is to add an additional linear term at each iteration, selected by solving a particular optimisation problem, for which primal and dual linear programming formulations are given.     

On the general inverse problem of optimal control theory

The general inverse problem of optimal control is considered from a dynamic programming point of view. Necessary and sufficient conditions are developed which two integral criteria must satisfy if they are to yield the same optimal feedback law, the dynamics being fixed. Specializing to the linear-quadratic case, it is shown how the general results given here recapture previously obtained results for quadratic criteria with linear dynamics.Dedicated to R. Bellman     

Convergent Lagrangian heuristics for nonlinear minimum cost network flows

We consider the separable nonlinear and strictly convex single-commodity network flow problem (SSCNFP). We develop a computational scheme for generating a primal feasible solution from any Lagrangian dual vector; this is referred to as “early primal recovery”. It is motivated by the desire to obtain a primal feasible vector before convergence of a Lagrangian scheme; such a vector is not available from a Lagrangian dual vector unless it is optimal. The scheme is constructed such that if we apply it from a sequence of Lagrangian dual vectors that converge to an optimal one, then the resulting primal (feasible) vectors converge to the unique optimal primal flow vector. It is therefore also a convergent Lagrangian heuristic, akin to those primarily devised within the field of combinatorial optimization but with the contrasting and striking advantage that it is guaranteed to yield a primal optimal solution in the limit. Thereby we also gain access to a new stopping criterion for any Lagrangian dual algorithm for the problem, which is of interest in particular if the SSCNFP arises as a subproblem in a more complex model. We construct instances of convergent Lagrangian heuristics that are based on graph searches within the residual graph, and therefore are efficiently implementable; in particular we consider two shortest path based heuristics that are based on the optimality conditions of the original problem. Numerical experiments report on the relative efficiency and accuracy of the various schemes.     

The continuity of the optimum in parametric programming and applications to stochastic programming

It is proved a sufficient condition that the optimal value of a linear program be a continuous function of the coefficients. The condition isessential, in the sense that, if it is not imposed, then examples with discontinuous optimal-value function may be found. It is shown that certain classes of linear programs important in applications satisfy this condition. Using the relation between parametric linear programming and the distribution problem in stochastic programming, a necessary and sufficient condition is given that such a program has optimal value. Stable stochastic linear programs are introduced, and a sufficient condition of such stability, important in computation problems, is established.This note is a slightly modified version of a paper presented at the Institute of Econometrics and Operations Research of the University of Bonn, Bonn, Germany, 1972.The author is grateful to G. B. Dantzig and S. Karamardian for useful comments on an earlier draft of this paper. In particular, S. Karamardian proposed modifications which made clearer the proof of Lemma 2.1.     

Sufficient conditions for total ill-posedness in linear semi-infinite optimization

This paper deals with the so-called total ill-posedness of linear optimization problems with an arbitrary (possibly infinite) number of constraints. We say that the nominal problem is totally ill-posed if it exhibits the highest unstability in the sense that arbitrarily small perturbations of the problem’s coefficients may provide both, consistent (with feasible solutions) and inconsistent problems, as well as bounded (with finite optimal value) and unbounded problems, and also solvable (with optimal solutions) and unsolvable problems. In this paper we provide sufficient conditions for the total ill-posedness property exclusively in terms of the coefficients of the nominal problem.     

Optimal Prioritized Infeasibility Handling in Model Predictive Control: Parametric Preemptive Multiobjective Linear Programming Approach

All practical implementations of model-based predictive control (MPC) require a means to recover from infeasibility. We propose a strategy designed for linear state-space MPC with prioritized constraints. It relaxes optimally an infeasible MPC optimization problem into a feasible one by solving a single-objective linear program (LP) online in addition to the standard online MPC optimization problem at each sample. By optimal, it is meant that the violation of a lower prioritized constraint cannot be made less without increasing the violation of a higher prioritized constraint. The problem of computing optimal constraint violations is naturally formulated as a parametric preemptive multiobjective LP. By extending well-known results from parametric LP, the preemptive multiobjective LP is reformulated into an equivalent standard single-objective LP. An efficient algorithm for offline design of this LP is given, and the algorithm is illustrated on an example.     

A simplified recurrent neural network for pseudoconvex optimization subject to linear equality constraints

In this paper, the optimization techniques for solving pseudoconvex optimization problems are investigated. A simplified recurrent neural network is proposed according to the optimization problem. We prove that the optimal solution of the optimization problem is just the equilibrium point of the neural network, and vice versa if the equilibrium point satisfies the linear constraints. The proposed neural network is proven to be globally stable in the sense of Lyapunov and convergent to an exact optimal solution of the optimization problem. A numerical simulation is given to illustrate the global convergence of the neural network. Applications in business and chemistry are given to demonstrate the effectiveness of the neural network.     

Multi-item capacitated lot-sizing by a Cross decomposition based algorithm

Multi-item capacitated lot-sizing involves the determination of production quantities in different periods in the presence of resource constraints. The model presented in this paper includes a setup time for the production of a lot of an item. The production of items in a given period is constrained by a limited regular time and a limited overtime. Moreover, the production level of any item in a given period is also limited. This problem is tackled by a Cross decomposition based algorithm which can provide an optimal solution, or a near optimal solution if computational time is restricted.     

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.     

Uniform quasi-concavity in probabilistic constrained stochastic programming

A probabilistic constrained stochastic linear programming problem is considered, where the rows of the random technology matrix are independent and normally distributed. The quasi-concavity of the constraining function needed for the convexity of the problem is ensured if the factors of the function are uniformly quasi-concave. A necessary and sufficient condition is given for that property to hold. It is also shown, through numerical examples, that such a special problem still has practical application in optimal portfolio construction.     

Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques

Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.     

A modified method for constructing efficient solutions structure of MOLP

This paper deals with a recently proposed algorithm for obtaining all weak efficient and efficient solutions in a multi objective linear programming (MOLP) problem. The algorithm is based on solving some weighted sum problems, and presents an easy and clear solution structure. We first present an example to show that the algorithm may fail when at least one of these weighted sum problems has not a finite optimal solution. Then, the algorithm is modified to overcome this problem. The modified algorithm determines whether an efficient solution exists for a given MOLP and generates the solution set correctly (if exists) without any change in the complexity.     

On alternative optimal solutions to quasimonotonic programming with linear constraints

In this paper, the nonlinear programming problem with quasimonotonic （ both quasiconvex and quasiconcave ）objective function and linear constraints is considered. With the decomposition theorem of polyhedral sets, the structure of optimal solution set for the programming problem is depicted. Based on a simplified version of the convex simplex method, the uniqueness condition of optimal solution and the computational procedures to determine all optimal solutions are given, if the uniqueness condition is not satisfied. An illustrative example is also presented.