Understanding linear semi-infinite programming via linear programming over cones

In this article, the standard primal and dual linear semi-infinite programming (DLSIP) problems are reformulated as linear programming (LP) problems over cones. Therefore, the dual formulation via the minimal cone approach, which results in zero duality gap for the primal–dual pair for LP problems over cones, can be applied to linear semi-infinite programming (LSIP) problems. Results on the geometry of the set of the feasible solutions for the primal LSIP problem and the optimality criteria for the DLSIP problem are also discussed.     

On the structure and properties of a linear multilevel programming problem

Many decision-making situations involve multiple planners with different, and sometimes conflicting, objective functions. One type of model that has been suggested to represent such situations is the linear multilevel programming problem. However, it appears that theoretical and algorithmic results for linear multilevel programming have been limited, to date, to the bounded case or the case of when only two levels exist. In this paper, we investigate the structure and properties of a linear multilevel programming problem that may be unbounded. We study the geometry of the problem and its feasible region. We also give necessary and sufficient conditions for the problem to be unbounded, and we show how the problem is related to a certain parametric concave minimization problem. The algorithmic implications of the results are also discussed.This research was supported by National Science Foundation Grant No. ECS-85-15231.     

Constructing efficient solutions structure of multiobjective linear programming

It is not a difficult task to find a weak Pareto or Pareto solution in a multiobjective linear programming (MOLP) problem. The difficulty lies in finding all these solutions and representing their structure. This paper develops an algorithm for solving this problem. We investigate the solutions and their relationships in the objective space. The algorithm determines finite number of weights, each of which corresponds to a weighted sum problems. By solving these problems, we further obtain all weak Pareto and Pareto solutions of the MOLP and their structure in the constraint space. The algorithm avoids the degeneration problem, which is a major hurdle of previous works, and presents an easy and clear solution structure.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

Multiple objective linear programming models with interval coefficients – an illustrated overview

In most real-world situations, the coefficients of decision support models are not exactly known. In this context, it is convenient to consider an extension of traditional mathematical programming models incorporating their intrinsic uncertainty, without assuming the exactness of the model coefficients. Interval programming is one of the tools to tackle uncertainty in mathematical programming models. Moreover, most real-world problems inherently impose the need to consider multiple, conflicting and incommensurate objective functions. This paper provides an illustrated overview of the state of the art of Interval Programming in the context of multiple objective linear programming models.     

NP-hardness of linear multiplicative programming and related problems

The linear multiplicative programming problem minimizes a product of two (positive) variables subject to linear inequality constraints. In this paper, we show NP-hardness of linear multiplicative programming problems and related problems.     

Numerical solution of positive control problem via linear programming

The reachability problem for linear time-invariant discrete-time control systems with sign-restricted input is considered. The time-optimal control is constructed by an iterative procedure. Each step of the iteration is defined as a linear programming problem. This problem is solved by the simplex algorithm. The initial feasible solution for the simplex algorithm is provided by the preceding step of the iteration. The inversion of the basis matrix is reduced to a bordering procedure. The structural stability of the solution is investigated.     

Complexity of linear programming

The complexity of linear programming is discussed in the “integer” and “real number” models of computation. Even though the integer model is widely used in theoretical computer science, the real number model is more useful for estimating an algorithm's running time in actual computation.Although the ellipsoid algorithm is a polynomial-time algorithm in the integer model, we prove that it has unbounded complexity in the real number model. We conjecture that there exists no polynomial-time algorithm for the linear inequalities problem in the real number model. We also conjecture that linear inequalities are strictly harder than linear equalities in all “reasonable” models of computation.     

A contribution to the linear programming approach to joint cost allocation: Methodology and application

The linear programming (LP) approach has been commonly proposed for joint cost allocation purposes. Within a LP framework, the allocation rules are based on a marginal analysis. Unfortunately, the additivity property which is required to completely allocate joint costs fails in presence of capacity, institutional or environmental constraints.     

Stability of the primal-dual partition in linear semi-infinite programming

We analyse the primal-dual states in linear semi-infinite programming (LSIP), where we consider the primal problem and the so called Haar's dual problem. Any linear programming problem and its dual can be classified as bounded, unbounded or inconsistent, giving rise to nine possible primal-dual states, which are reduced to six by the weak duality property. Recently, Goberna and Todorov have studied this partition and its stability in continuous LSIP in a series of papers [M.A. Goberna and M.I. Todorov, Primal, dual and primal-dual partitions in continuous linear semi-infinite programming, Optimization 56 (2007), pp. 617–628; M.A. Goberna and M.I. Todorov, Generic primal-dual solvability in continuous linear semi-infinite programming, Optimization 57 (2008), pp. 239–248]. In this article we consider the general case, with no continuity assumptions, discussing the maintenance of the primal-dual state of the problem by allowing small perturbations of the data. We characterize the stability of all of the six possible primal-dual states through necessary and sufficient conditions which depend on the data, and can be easily checked, showing some differences with the continuous case. These conditions involve the strong Slater constraint qualification, and some distinguished convex sets associated to the data.     

Checking solvability of systems of interval linear equations and inequalities via mixed integer programming

This paper deals with the problems of checking strong solvability and feasibility of linear interval equations, checking weak solvability of linear interval equations and inequalities, and finding control solutions of linear interval equations. These problems are known to be NP$\mathit{NP}$-hard. We use some recently developed characterizations in combination with classical arguments to show that these problems can be equivalently stated as optimization tasks and provide the corresponding linear mixed 0–1 programming formulations.     

A continuous approach for the concave cost supply problem via DC programming and DCA

The paper addresses an important but difficult class of concave cost supply management problems which consist in minimizing a separable increasing concave objective function subject to linear and disjunctive constraints. We first recast these problems into mixed zero-one nondifferentiable concave minimization over linear constraints problems and then apply exact penalty techniques to state equivalent nondifferentiable polyhedral DC (Difference of Convex functions) programs. A new deterministic approach based on DC programming and DCA (DC Algorithms) is investigated to solve the latter ones. Finally numerical simulations are reported which show the efficiency, the robustness and the globality of our approach.     

Designing a majorization scheme for the recourse function in two-stage stochastic linear programming

We discuss issues pertaining to the domination from above of the second-stage recourse function of a stochastic linear program and we present a scheme to majorize this function using a simpler sublinear function. This majorization is constructed using special geometrical attributes of the recourse function. The result is a proper, simplicial function with a simple characterization which is well-suited for calculations of its expectation as required in the computation of stochastic programs. Experiments indicate that the majorizing function is well-behaved and stable.     

A dual approach to solve the fuzzy linear programming problem

A concept of fuzzy objective based on the Fuzzification Principle is presented. In accordance with this concept, the Fuzzy Linear Mathematical Programming problem is easily solved. A relationship of duality among fuzzy constraints and fuzzy objectives is given. The dual problem of a Fuzzy Linear Programming problem is also defined.     

Cutting plane algorithms for the inverse mixed integer linear programming problem

We present cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP), which is to minimally perturb the objective function of a mixed integer linear program in order to make a given feasible solution optimal.     

A compromise procedure for the multiple objective linear fractional programming problem

A generalization of a well-known multiple objective linear fractional programming (MOLFP) problem, the multiple objective fractional programming (MOFP) problem, is formulated. A concept of multiple objective programming (MOP) problem corresponding to MOFP is introduced and some relations between those problems are examined. Based on these results, a compromise procedure for MOLFP problem is proposed. A numerical example is given to show how the procedure works.     

Asymptotic behaviour of Karmarkar's method for linear programming

The asymptotic behaviour of Karmarkar's method is studied and an estimate of the rate of the objective function value decrease is given. Two possible sources of numerical instability are discussed and a stabilizing procedure is proposed.Research supported in part by Republicka zajednica za nauku SR Srbije.     

Determination of the efficient set in multiobjective linear programming

This paper develops a method for finding the whole set of efficient points of a multiobjective linear problem. Two algorithms are presented; the first one describes the set of all efficient vertices and all efficient rays of the constraint polyhedron, while the second one generates the set of all efficient faces. The method has been tested on several examples for which numerical results are reported.The authors are grateful to Professor W. Stadler and an anonymous referee for their helpful comments and corrections.     

Some properties of generalized proximal point methods for quadratic and linear programming

The proximal point method for convex optimization has been extended recently through the use of generalized distances (e.g., Bregman distances) instead of the Euclidean one. One advantage of these extensions is the possibility of eliminating certain constraints (mainly positivity) from the subproblems, transforming an inequality constrained problem into a sequence of unconstrained or equality constrained problems. We consider here methods obtained using a certain class of Bregman functions applied to convex quadratic (including linear) programming, which are of the interior-point type. We prove that the limit of the sequence generated by the method lies in the relative interior of the solution set, and furthermore is the closest optimal solution to the initial point, in the sense of the Bregman distance. These results do not hold for the standard proximal point method, i.e., when the square of the Euclidean norm is used as the Bregman distance.The research leading to this paper was partially supported by CNPq Grant 301280/86.We thank two anonymous referees whose comments and suggestions allowed us to improve our results in a significant way.