Maximal and supremal tolerances in multiobjective linear programming

Let a multiobjective linear programming problem and any efficient solution be given. Tolerance analysis aims to compute interval tolerances for (possibly all) objective function coefficients such that the efficient solution remains efficient for any perturbation of the coefficients within the computed intervals. The known methods either yield tolerances that are not the maximal possible ones, or they consider perturbations of weights of the weighted sum scalarization only. We focus directly on perturbations of the objective function coefficients, which makes the approach independent on a scalarization technique used. In this paper, we propose a method for calculating the supremal tolerance (the maximal one need not exist). The main disadvantage of the method is the exponential running time in the worst case. Nevertheless, we show that the problem of determining the maximal/supremal tolerance is NP-hard, so an efficient (polynomial time) procedure is not likely to exist. We illustrate our approach on examples and present an application in transportation problems. Since the maximal tolerance may be small, we extend the notion to individual lower and upper tolerances for each objective function coefficient. An algorithm for computing maximal individual tolerances is proposed.     

Nondifferentiable mathematical programming and convex-concave functions

For a convex-concave functionL(x, y), we define the functionf(x) which is obtained by maximizingL with respect toy over a specified set. The minimization problem with objective functionf is considered. We derive necessary conditions of optimality for this problem. Based upon these necessary conditions, we define its dual problem. Furthermore, a duality theorem and a converse duality theorem are obtained. It is made clear that these results are extensions of those derived in studies on a class of nondifferentiable mathematical programming problems.This work was supported by the Japan Society for the Promotion of Sciences.     

Nonlinear one-parametric bottleneck linear programming

We consider the one-parametric linear bottleneck problem min {c (x, t) ¦ x P (t)} where the bottleneck objectivec (x, t):=max {c_j (t) ¦ x_j>0} is minimized subject to linear constraints, i.e. P(t):={x ¦A (t) ·x=b (t), x0}. All coefficients are assumed to be continuous functions of one real parametert which varies in a real intervalS. A method is developed for constructing a partition ofS into subintervals on which either a basis stays optimal or the problem stays infeasible. Finiteness of the partition is due to certain finiteness assumptions on the zeroes of particular combinations of the coefficient functions. Using a lexicographic refinement of the objective function a characterization of the optimality interval of a fixed basis is derived which is independent on explicit information about other bases.

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Zusammenfassung Es werden einparametrische lineare Engpaßprobleme min {c (x, t) ¦x P (t)} betrachtet, wobei die Engpaßzielfunktionc (x, t):=max {c_j (t) ¦ x_j >0} unter linearen Nebenbedingungen, die durch P(t):={x ¦A (t) ·x=b (t), x0} gegeben sind, minimiert wird. Dabei wird angenommen, daß alle Koeffizienten stetige Funktionen eines reellen Parameterst aus einem IntervallS R sind. Es wird eine Methode entwickelt,S derart in Teilintervalle zu zerlegen, daß entweder eine Basis in solch einem Teilintervall optimal oder das Problem unzulässig bleibt. Die Endlichkeit dieser Partition vonS ergibt sich aus Endlichkeitsvoraussetzungen für die Nullstellen von Funktionen, die sich als gewisse Kombinationen der Koeffizientenfunktionen schreiben lassen. Durch eine lexikographische Verfeinerung der Zielfunktion gelingt es, das Intervall zu charakterisieren, in dem eine feste Basis optimal ist. Diese Charakterisierung ist unabhängig von expliziten Informationen über andere Basen.

     

Bottleneck linear programming

The Bottleneck Linear Programming problem (BLP) is to maximizex ₀ = max _j c _j ,x _j > 0 subject toAx = b, x 0. A relationship between the BLP and a problem solvable by a greedy algorithm is established. Two algorithms for the BLP are developed and computational experience is reported.     

An approximation algorithm for convex multi-objective programming problems

In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson’s outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.     

Enclosing solutions of complex linear systems of equations iteratively

We consider the interval iteration [x ]^{k +1} = [A ][x ]^k + [b ] in different interval arithmetics with the aim to enclose solutions of x = Ax +b in the case that A and b are only known to be contained in some given intervals. We give necessary and sufficient criteria for the convergence of the interval iteration for every initial interval vector [x ]⁰ to some [x ]* = [x ]*([x ]⁰) with respect to the considered interval arithmetic. Such a limit is a solution of the interval system [x ] = [A ][x ] + [b ]. If we compare the interval arithmetics with respect to the behavior of [x ]^{k +1} = [A ][x ]^k + [b ] we come to the conclusion, that the special choice of the arithmetic has a sensitive influence on the convergence of the sequence. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A note on duality gaps in linear programming over convex sets

For certain types of mathematical programming problems, a related dual problem can be constructed in which the objective value of the dual problem is equal to the objective function of the given problem. If these two problems do not have equal values, a duality gap is said to exist. No such gap exists for pairs of ordinary dual linear programming problems, but this is not the case for linear programming problems in which the nonnegativity conditionx ? 0 is replaced by the condition thatx lies in a certain convex setK. Duffin (Ref. 1) has shown that, whenK is a cone and a certain interiority condition is fulfilled, there will be no duality gap. In this note, we show that no duality gap exists when the interiority condition is satisfied andK is an arbitrary closed convex set inR ⁿ .     

The almost chebyshev property in linear semi-infinite programming

We consider a linear semi-infinite programming problem where the index set of the constraints is compact and the constraint functions are continuous on it. The set of all continuous functions on this index set as right hand sides are the parameter set. We investigate how large various unicity sets are.We state a condition on the objective function vector and the “matrix” of the problem which characterizes when the set of a parameters with a non-unique optimal point is a set of the first Baire category in the solvability set. This is the case if and only if the unicity set is a dense subset of the solvability set. Under the same assumptions it is even true that the interior of the strong unicity set is I also dense. If the index set of the constraints contains a dense subset with the property that each point1 is a G ₈-set, then the parameters of the strong unicity set, such that the optimal point satisfies the linear independence constraint qualification, are also dense.

We apply our results to a characterization of a unique continuous selection for the optimal set I mapping and to a one-sided L ₁-approximation problem     

A feasible direction method for linear programming

We discuss a finite method of a feasible direction for linear programming problems. The method begins with a feasible basic vector for the problem, constructs a profitable direction to move using the updated column vectors of the nonbasic variables eligible to enter this basic vector. It then moves in this direction as far as possible, while retaining feasibility. This move in general takes it though the relative interior of a face of th set of a feasible solutions. The final point, $\text{x}$, obtained at the end of this move will not in general be a basic solution. Using $\text{x}$ the method then constructs a basic feasible solution at which the objective value is better than, or the same as that at $\text{x}$. The whole process repeats with the new basic feasible solution. We show that this method can be implemented using basis inverses. Initial computer runs of this method in comparison with the usual edge following primary simplex algorithms are very encouraging.     

On the complexity of minmax regret linear programming

We consider linear programming problems with uncertain objective function coefficients. For each coefficient of the objective function, an interval of uncertainty is known, and it is assumed that any coefficient can take on any value from the corresponding interval of uncertainty, regardless of the values taken by other coefficients. It is required to find a minmax regret solution. This problem received considerable attention in the recent literature, but its computational complexity status remained unknown. We prove that the problem is strongly NP-hard. This gives the first known example of a minmax regret optimization problem that is NP-hard in the case of interval-data representation of uncertainty but is polynomially solvable in the case of discrete-scenario representation of uncertainty.     

A new condition number for linear programming

In this paper we define a new condition number ?(A) for the following problem: given a m by n matrix A, find x∈ℝ ⁿ , s.t. Ax<0. We characterize this condition number in terms of distance to ill-posedness and we compare it with existing condition numbers for the same problem. Received: November 5, 1999 / Accepted: November 2000?Published online September 17, 2001     

An extension of the simplex algorithm for semi-infinite linear programming

We present a primal method for the solution of the semi-infinite linear programming problem with constraint index setS. We begin with a detailed treatment of the case whenS is a closed line interval in . A characterization of the extreme points of the feasible set is given, together with a purification algorithm which constructs an extreme point from any initial feasible solution. The set of points inS where the constraints are active is crucial to the development we give. In the non-degenerate case, the descent step for the new algorithm takes one of two forms: either an active point is dropped, or an active point is perturbed to the left or right. We also discuss the form of the algorithm when the extreme point solution is degenerate, and in the general case when the constraint index set lies in ^p . The method has associated with it some numerical difficulties which are at present unresolved. Hence it is primarily of interest in the theoretical context of infinite-dimensional extensions of the simplex algorithm.     

Analysis of the objective space in multiple objective linear programming

A multiple objective linear program is defined by a matrix C consisting of the coefficients of the linear objectives and a convex polytope X defined by the linear constraints. An analysis of the objective space Y = C[X] for this problem is presented. A characterization between a face of Y and the corresponding faces of X is obtained. This result gives a necessary and sufficient condition for a face to be efficient. The theory and examples demonstrate the collapsing (simplification) that occurs in mapping X to Y. These results form a basis for a new approach to analyzing multiple objective linear programs.     

A linear integer programming bound for maximum-entropy sampling

 We introduce a new upper bound for the maximum-entropy sampling problem. Our bound is described as the solution of a linear integer program. The bound depends on a partition of the underlying set of random variables. For the case in which each block of the partition has bounded cardinality, we describe an efficient dynamic-programming algorithm to calculate the bound. For the very special case in which the blocks have no more than two elements, we describe an efficient algorithm for calculating the bound based on maximum-weight matching. This latter formulation has particular value for local-search procedures that seek to find a good partition. We relate our bound to recent bounds of Hoffman, Lee and Williams. Finally, we report on the results of some computational experiments. Received: September 27, 2000 / Accepted: July 26, 2001 Published online: September 5, 2002 Key words. experimental design – design of experiments – entropy – maximum-entropy sampling – matching – integer program – spectral bound – Fischer's inequality – branch-and-bound – dynamic programming Mathematics Subject Classification (2000): 52B12, 90C10 Send offprint requests to: Jon Lee Correspondence to: Jon Lee     

On strong optimality of interval linear programming

We consider a linear programming problem with interval data. We discuss the problem of checking whether a given solution is optimal for each realization of interval data. This problem was studied for particular forms of linear programming problems. Herein, we extend the results to a general model and simplify the overall approach. Moreover, we inspect computational complexity, too. Eventually, we investigate a related optimality concept of semi-strong optimality, showing its characterization and complexity.     

An explicit linear solution for the quadratic dynamic programming problem

For a given vectorx ₀, the sequence {x _t} which optimizes the sum of discounted rewardsr(x _t, x_t+1), wherer is a quadratic function, is shown to be generated by a linear decision rulex _t+1=Sx _t +R. Moreover, the coefficientsR,S are given by explicit formulas in terms of the coefficients of the reward functionr. A unique steady-state is shown to exist (except for a degenerate case), and its stability is discussed.     

Using objective values to start multiple objective linear programming algorithms

We introduce in this paper a new starting mechanism for multiple-objective linear programming (MOLP) algorithms. This makes it possible to start an algorithm from any solution in objective space. The original problem is first augmented in such a way that a given starting solution is feasible. The augmentation is explicitly or implicitly controlled by one parameter during the search process, which verifies the feasibility (efficiency) of the final solution. This starting mechanism can be applied either to traditional algorithms, which search the exterior of the constraint polytope, or to algorithms moving through the interior of the constraints. We provide recommendations on the suitability of an algorithm for the various locations of a starting point in objective space. Numerical considerations illustrate these ideas.     

Finding normal solutions in piecewise linear programming

Letf: ⁿ (–, ] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg min _x ¦x¦²/2+t f(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)} _t 0, minimizingf over an ellipsoid, and choosing penalty parameters inL ₁QP methods for strictly convex QP.This research was supported by the State Committee for Scientific Research under Grant 8S50502206.     

Some NP-complete problems in linear programming

Degeneracy checking in linear programming is NP-complete. So is the problem of checking whether there exists a basic feasible solution with a specified objective value.