The set of all nondominated solutions in linear cases and a multicriteria simplex method

In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.     

A note on efficient solutions for the linear bilevel programming problem

The solutions of linear bilevel programming problems frequently are non-Pareto-optimal. The potential increase in payoffs generated by Pareto improvements makes it worthwhile to consider methods with which to move the solution to the efficient frontier. Bargaining models offer one class of solutions, which, contrary to the original non-cooperative, sequential decision-making situation, however, assume cooperation. We make an attempt to maintain the original power structure by introducing the asymmetric Nash bargaining solution.     

A method for solving the general parametric linear complementarity problem

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.     

Optimization of a linear function over the set of stochastic efficient solutions

In this paper we study the problem of optimization over an integer efficient set of a Multiple Objective Integer Linear Stochastic Programming problem. Once the problem is converted into a deterministic one by adapting the $2$ -levels recourse approach, a new pivoting technique is applied to generate an optimal efficient solution without having to enumerate all of them. This method combines both techniques, the L-shaped method and the combined method developed in Chaabane and Pirlot (J Ind Manag Optim 6:811–823, 2010). A detailed didactic example is given to illustrate different steps of our algorithm.     

A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program

Various computational difficulties arise in using decision set-based vector maximization methods to solve multiple objective linear programming problems. As a result, several researchers have begun to explore the possibility of solving these problems by examining subsets of their outcome sets, rather than of their decision sets. In this article, we present and validate a basic weight set decomposition approach for generating the set of all efficient extreme points in the outcome set of a multiple objective linear program. Based upon this approach, we then develop an algorithm, called the Weight Set Decomposition Algorithm, for generating this set. A sample problem is solved using this algorithm, and the main potential computational and practical advantages of the algorithm are indicated.     

Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem

This article presents a finite, outcome-based algorithm for optimizing a lower semicontinuous function over the efficient set of a bicriteria linear programming problem. The algorithm searches the efficient faces of the outcome set of the bicriteria linear programming problem. It exploits the fact that the dimension of the outcome set of the bicriteria problem is at most two. As a result, in comparison to decisionbased approaches, the number of efficient faces that need to be found is markedly reduced. Furthermore, the dimensions of the efficient faces found are always at most one. The algorithm can be implemented for a wide variety of lower semicontinuous objective functions.     

A general method for enclosing solutions of interval linear equations

We describe a general method for enclosing the solution set of a system of interval linear equations. We present a general theorem and an algorithm in a MATLAB-style code.     

A polynomial-time method to compute all Nash equilibria solutions of a general two-person inspection game

We consider a two-person nonzero-sum simultaneous inspection game that takes place at multiple sites. The inspector has a limited inspection resource. She needs to decide which sites to inspect, and with how much effort, while adhering also to local restrictions on the permitted inspections levels at the sites. The inspectee has several employees who work on his behalf. He needs to decide how to distribute them across the sites, and how they should act there. Computation of Nash equilibria is challenging for this sort of games. Still, we develop a linear-time algorithm that determines all Nash equilibria solutions of the game, and provide explicit (easily computable) expressions for all possible Nash equilibria. We then derive some managerial insights by applying the algorithm to several examples, and examining the Nash equilibria, including an outcome that an increase in the inspection resource may induce the inspectee to cooperate more at sites without increasing the inspection levels at them.     

Ellipsoids that contain all the solutions of a positive semi-definite linear complementarity problem

This paper deals with the LCP (linear complementarity problem) with a positive semi-definite matrix. Assuming that a strictly positive feasible solution of the LCP is available, we propose ellipsoids each of which contains all the solutions of the LCP. We use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the LCP. We can apply the lower bound to test whether a given variable is positive over the solution set of the LCP. That is, if the lower bound is positive, we know that the variable is positive over the solution set of the LCP; hence, by the complementarity condition, its complement is zero. In this case we can eliminate the variable and its complement from the LCP. We also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path-following algorithm proposed by the authors for the LCP. If the LCP has a unique non-degenerate solution, the lower bound and the upper bound for the solution, computed at each iteration of the path-following algorithm, both converge to the solution of the LCP.Supported by Grant-in-Aids for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.Supported by Grant-in-Aids for Young Scientists (63730014) and for General Scientific Research (63490010) of The Ministry of Education, Science and Culture.     

Algorithm for determining the reachability set of a linear control system

We present in this paper algorithms for calculating the reachability set of a linear control system with a bounded closed control set and a finite time interval. We also present algorithms for the time-optimal problem of the linear control that yields an approximation to the optimal time and the corresponding control function. We give numerical examples of the computer implementation of these algorithms.This research was supported by the National Science Foundation of China.The authors are grateful to an anonymous referee for constructive suggestions that greatly helped to improve the presentation.     

Finding the projection of a given point on the set of solutions of a linear programming problem

The problem of finding the projections of points on the sets of solutions of primal and dual problems of linear programming is considered. This problem is reduced to a single solution of the problem of minimizing a new auxiliary function, starting from some threshold value of the penalty coefficient. Estimates of the threshold value are obtained. A software implementation of the proposed method is compared with some known commercial and research software packages for solving linear programming problems.     

A cell exclusion algorithm for determining all the solutions of a nonlinear system of equations

A new group of methods named cell exclusion algorithms (CEAs) is developed for finding all the solutions of a nonlinear system of equations. These types of algorithms, different in principle from those of homotopy, interval and cell-mapping-dynamical-analysis approaches, are based on cellular discretization and the use of a certain simple necessity test of the solutions. The main advantages of the algorithms are their simplicity, reliability, and general applicability. Having all features of interval techniques (but without using interval arithmetic) and with complexity O(log(1/)), the algorithms improve significantly on both the interval algorithms and the cell mapping techniques. Theoretical analysis and numerical simulations both demonstrate that CEAs are very efficient.     

Optimizing a linear function over an efficient set

The problem (P) of optimizing a linear functiond ^T x over the efficient set for a multiple-objective linear program (M) is difficult because the efficient set is typically nonconvex. Given the objective function directiond and the set of domination directionsD, ifd ^T 0 for all nonzero D, then a technique for finding an optimal solution of (P) is presented in Section 2. Otherwise, given a current efficient point , if there is no adjacent efficient edge yielding an increase ind ^T x, then a cutting plane is used to obtain a multiple-objective linear program ( ) with a reduced feasible set and an efficient set . To find a better efficient point, we solve the problem (I_i) of maximizingc _i ^T x over the reduced feasible set in ( ) sequentially fori. If there is a that is an optimal solution of (I_i) for somei and , then we can choosex ⁱ as a current efficient point. Pivoting on the reduced feasible set allows us to find a better efficient point or to show that the current efficient point is optimal for (P). Two algorithms for solving (P) in a finite sequence of pivots are presented along with a numerical example.The authors would like to thank an anonymous referee, H. P. Benson, and P. L. Yu for numerous helpful comments on this paper.     

A method for solving a general multi-valued complementarity problem

We propose an extended version of Chandrasekaran’s method for general complementarity problems with multi-valued weakly off-diagonally antitone costmappings. It allows one either to construct a sequence converging to a solution or to recognize that the problem has no solutions. We also propose versions of Jacobi’s methods for multi-valued inclusions subject to one- and two-side constraints.     

Estimates of solutions of a general initial-boundary-value problem for the linear nonstationary system of Navier-Stokes equations in a half-space

In the half-spaceX ₃>0 one considers the initial-boundary-value problem for the Stokes system in which the boundary conditions are given by means of an arbitrary matricial differential operator of size 3×4. One proves that, under certain restrictions on this operator, the solution satisfies coercive estimates in the norms of W_ρ ^2?,?.     

Bounds for the singular set of solutions to non linear elliptic systems

We give new estimates for the Hausdorff dimension of the singular set of solutions to elliptic systems If the vector fields a and b are Hölder continuous with respect to the variables (x,u) with exponent , then, under suitable assumptions, the Hausdorff dimension of the singular set of any weak solution is at most . We consider natural growth assumptions on a(x,u,Du) with respect to u and critical ones on the right hand side b(x,u,Du), with respect to Du.Accepted: 12 March 2003, Published online: 16 May 2003     

An efficient algorithm for the complete set partitioning problem

The complete set partitioning problem is the well known set partitioning problem with all possible nonzero binary columns in the constraint matrix. A highly specialized enumerative algorithm, which never requires the explicit maintenance of the constraint matrix, is presented. Computational results, with data reflecting a particular corporate tax payment scenario that can be modelled as a complete set partitioning problem, is also given.