Optimality conditions for nondifferentiable minimax fractional programming with complex variables

We show that a minimax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. We establish the necessary and sufficient optimality conditions of nondifferentiable minimax fractional programming problem with complex variables under generalized convexities.     

Complex Minimax Fractional Programming of Analytic Functions

We prove that a minmax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. Using a parametric approach, we establish the Kuhn-Tucker type necessary optimality conditions and prove the existence theorem of optimality for complex minimax fractional programming in the framework of generalized convexity. Subsequently, we apply the optimality conditions to formulate a one-parameter dual problem and prove weak duality, strong duality, and strict converse duality theorems involving generalized convex complex functions. This research was partly supported by NSC, Taiwan.     

A simplex algorithm for piecewise-linear fractional programming problems

Generalizations of the well-known simplex method for linear programming are available to solve the piecewise linear programming problem and the linear fractional programming problem. In this paper we consider a further generalization of the simplex method to solve piecewise linear fractional programming problems unifying the simplex method for linear programs, piecewise linear programs, and the linear fractional programs. Computational results are presented to obtain further insights into the behavior of the algorithm on random test problems.     

Parametric simplex algorithms for solving a special class of nonconvex minimization problems

It is shown that parametric linear programming algorithms work efficiently for a class of nonconvex quadratic programming problems called generalized linear multiplicative programming problems, whose objective function is the sum of a linear function and a product of two linear functions. Also, it is shown that the global minimum of the sum of the two linear fractional functions over a polytope can be obtained by a similar algorithm. Our numerical experiments reveal that these problems can be solved in much the same computational time as that of solving associated linear programs. Furthermore, we will show that the same approach can be extended to a more general class of nonconvex quadratic programming problems.     

Global optimization for sum of geometric fractional functions

This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm.     

A simplex algorithm for piecewise-linear programming I: Derivation and proof

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming.Part I derives and justifies the essential steps of the algorithm, by extension from the simplex method for linear programming in bounded variables. The proof employs familiar finite-termination arguments and established piecewise-linear duality theory.Part II considers the relaxation of technical assumptions pertaining to finiteness, feasibility and nondegeneracy of piecewise-linear programs. Degeneracy is found to have broader consequences than in the linear case, and the standard techniques for prevention of cycling are extended accordingly.Part III analyzes the computational requirements of piecewise-linear programming. The direct approach embodied in the piecewise-linear simplex algorithm is shown to be inherently more efficient than indirect approaches that rely on transformation of piecewise-linear programs to equivalent linear programs. A concluding section surveys the many applications of piecewise-linear programming in linear programming,l ₁ estimation, goal programming, interval programming, and nonlinear optimization.This research has been supported in part by the National Science Foundation under grant MCS-8217261.     

Optimality in transient markov chains and linear programming

It is shown how a discrete Markov programming problem can be transformed, using a linear program, into an equivalent problem from which the optimal decision rule can be trivially deduced. This transformation is applied to problems which have either transient probabilities or discounted costs.This research was supported by the National Research Council of Canada, Grant A7751.     

Global optimization for sum of generalized fractional functions

This paper considers the solution of generalized fractional programming (GFP) problem which contains various variants such as a sum or product of a finite number of ratios of linear functions, polynomial fractional programming, generalized geometric programming, etc. over a polytope. For such problems, we present an efficient unified method. In this method, by utilizing a transformation and a two-part linearization method, a sequence of linear programming relaxations of the initial nonconvex programming problem are derived which are embedded in a branch-and-bound algorithm. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.     

Strong duality for robust minimax fractional programming problems

We develop a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. Following the framework of robust optimization, we establish strong duality between the robust counterpart of an uncertain minimax convex–concave fractional program, termed as robust minimax fractional program, and the optimistic counterpart of its uncertain conventional dual program, called optimistic dual. In the case of a robust minimax linear fractional program with scenario uncertainty in the numerator of the objective function, we show that the optimistic dual is a simple linear program when the constraint uncertainty is expressed as bounded intervals. We also show that the dual can be reformulated as a second-order cone programming problem when the constraint uncertainty is given by ellipsoids. In these cases, the optimistic dual problems are computationally tractable and their solutions can be validated in polynomial time. We further show that, for robust minimax linear fractional programs with interval uncertainty, the conventional dual of its robust counterpart and the optimistic dual are equivalent.     

Efficient algorithms for solving certain nonconvex programs dealing with the product of two affine fractional functions

Two algorithms for finding a global minimum of the product of two affine fractional functions over a compact convex set and solving linear fractional programs with an additional constraint defined by the product of two affine fractional functions are proposed. The algorithms are based on branch and bound techniques using an adaptive branching operation which takes place in one-dimensional intervals. Results from numerical experiments show that large scale problems can be efficiently solved by the proposed methods.     

Global optimization for the sum of generalized polynomial fractional functions

In this paper, a branch and bound approach is proposed for global optimization problem (P) of the sum of generalized polynomial fractional functions under generalized polynomial constraints, which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. By utilizing an equivalent problem and some linear underestimating approximations, a linear relaxation programming problem of the equivalent form is obtained. Consequently, the initial non-convex nonlinear problem (P) is reduced to a sequence of linear programming problems through successively refining the feasible region of linear relaxation problem. The proposed algorithm is convergent to the global minimum of the primal problem by means of the solutions to a series of linear programming problems. Numerical results show that the proposed algorithm is feasible and can successfully be used to solve the present problem (P).     

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.     

Transcendental geometric programs

This paper treats a class of posynomial-like functions whose variables may appear also as exponents or in logarithms. It is shown that the resulting programs, called transcendental geometric programs, retain many useful properties of ordinary geometric programs, although the new class of problems need not have unique minima and cannot, in general, be transformed into convex programs. A duality theory, analogous to geometric programming duality, is formulated under somewhat more restrictive conditions. The dual constraints are not all linear, but the notion ofdegrees of difficulty is maintained in its geometric programming sense. One formulation of the dual program is shown to be a generalization of the chemical equilibrium problem where correction factors are added to account for nonideality. Some of the computational difficulties in solving transcendental programs are discussed briefly.This research was partially supported by the National Institute of Health Grant No. GM-14789; Office of Naval Research under Contract No. N00014-75-C-0276; National Science Foundation Grant No. MPS-71-03341 A03; and the US Atomic Energy Commission Contract No. AT(04-3)-326 PA #18.     

Generating Sum-of-Ratios Test Problems in Global Optimization

A method is presented for the construction of test problems involving the minimization over convex sets of sums of ratios of affine functions. Given a nonempty, compact convex set, the method determines a function that is the sum of linear fractional functions and attains a global minimum over the set at a point that can be found by convex programming and univariate search. Generally, the function will have also local minima over the set that are not global minima.     

Robust Duality for Fractional Programming Problems with Constraint-Wise Data Uncertainty

In this paper, we examine duality for fractional programming problems in the face of data uncertainty within the framework of robust optimization. We establish strong duality between the robust counterpart of an uncertain convex–concave fractional program and the optimistic counterpart of its conventional Wolfe dual program with uncertain parameters. For linear fractional programming problems with constraint-wise interval uncertainty, we show that the dual of the robust counterpart is the optimistic counterpart in the sense that they are equivalent. Our results show that a worst-case solution of an uncertain fractional program (i.e., a solution of its robust counterpart) can be obtained by solving a single deterministic dual program. In the case of a linear fractional programming problem with interval uncertainty, such solutions can be found by solving a simple linear program.     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

A simplex algorithm for piecewise-linear programming III: Computational analysis and applications

The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program. The advantages are shown to be implicit in relationships between the linear and piecewise-linear algorithms, and to be independent of many details of implementation. Two sets of computational results serve to illustarate these arguments; the piecewise-linear simplex algorithm is observed to run 2–6 times faster than a comparable linear algorithm, not including any additional expense that might be incurred in setting up the equivalent linear program. Further support for the practical value of a good piecewise-linear programming algorithm is provided by a survey of many varied applications.This research has been supported in part by the National Science Foundation under grant DMS-8217261.     

On the structure of convex piecewise quadratic functions

Convex piecewise quadratic functions (CPQF) play an important role in mathematical programming, and yet their structure has not been fully studied. In this paper, these functions are categorized into difference-definite and difference-indefinite types. We show that, for either type, the expressions of a CPQF on neighboring polyhedra in its domain can differ only by a quadratic function related to the common boundary of the polyhedra. Specifically, we prove that the monitoring function in extended linear-quadratic programming is difference-definite. We then study the case where the domain of the difference-definite CPQF is a union of boxes, which arises in many applications. We prove that any such function must be a sum of a convex quadratic function and a separable CPQF. Hence, their minimization problems can be reformulated as monotropic piecewise quadratic programs.This research was supported by Grant DDM-87-21709 of the National Science Foundation.     

Solution of and bounding in a linearly constrained optimization problem with convex,polyhedral objective function

A dual method is presented to solve a linearly constrained optimization problem with convex, polyhedral objective function, along with a fast bounding technique, for the optimum value. The method can be used to solve problems, obtained from LPs, where some of the constraints are not required to be exactly satisfied but are penalized by piecewise linear functions, which are added to the objective function of the original problem. The method generalizes an earlier solution technique developed by Prékopa (1990). Applications to stochastic programming are also presented.This research was supported by the National Science Foundation, Grant No. DMS-9005159.Corresponding author.     

Linear Approximations in a Dynamic Programming Approach for the Uncapacitated Single-Source Minimum Concave Cost Network Flow Problem in Acyclic Networks

We consider minimum concave cost flow problems in acyclic, uncapacitated networks with a single source. For these problems a dynamic programming scheme is developed. It is shown that the concave cost functions on the arcs can be approximated by linear functions. Thus the considered problem can be solved by a series of linear programs. This approximation method, whose convergence is shown, works particularly well, if the nodes of the network have small degrees. Computational results on several classes of networks are reported.