Lipschitz selections and stability for quasiconvex programs

Stability of ε-optimal solutions for quasiconvex programs is studied.     

Duality for minmax programs

A duality theory is derived for minimizing the maximum of a finite set of convex functions subject to a convex constraint set generated by both linear and nonlinear inequalities. The development uses the theory of generalised geometric programming. Further, a particular class of minmax program which has some practical significance is considered and a particularly simple dual program is obtained.     

Duality for nonlinear fractional programs

Summary This paper develops duality results for nonlinear fractional programming problems. This is accomplished by using some known results connecting the solutions of a nonlinear fractional program with those of a suitably defined parametric convex program.     

Duality methods for a class of quasilinear systems

Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge–Bäcklund transformation, underlying symmetries among superficially different forms of the equations.     

Beyond polyconvexity: an existence result for a class of quasiconvex hyperelastic materials

This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non‐interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split‐quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.     

Duality for quasi-concave programs with explicit constraints

The idea of duality is now well established in the theory of concave programming. The basis of this duality is the concave conjugate transform. This has been exemplified in the development of generalised geometric programming. Much of the current research in duality theory is focused on relaxing the requirement of concavity. Here we develop a duality theory for mathematical programs with a quasi concave objective function and explicit quasi concave constraints. Generalisations of the concave conjugate transform are introduced which pair quasi concave functions as the concave conjugate transform does for concave functions. Optimality conditions are derived relating the primal quasi concave program to its dual. This duality theory was motivated by and has implications in certain problems of mathematical economics. An application to economics is given.     

Duality theorems for a class of nonconvex programming problems

The object of this paper is to prove duality theorems for quasiconvex programming problems. The principal tool used is the transformation introduced by Manas for reducing a nonconvex programming problem to a convex programming problem. Duality in the case of linear, quadratic, and linear-fractional programming is a particular case of this general case.The authors are grateful to the referees for their kind suggestions.     

Duality for a class of nondifferentiable mathematical programming problems

Two dual problems are proposed for the minimax problem: minimize max_y?Yφ(x, y), subject to g(x) ? 0. A duality theorem is established for each dual problem. It is revealed that these problems are intimately related to a class of nondifferentiable programming problems.     

Duality for a class of continuous-time homogeneous fractional programming problems

In this note we present a continuous-time analogue of a duality formulation due to Craven and Mond for a class of homogeneous fractional programming problems. In this duality formulation, the dual problem is also a fractional program with the same objective function as the primal problem.

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Zusammenfassung In dieser Arbeit wird der Dualitätsansatz von Craven und Mond für homogene Quotientenprogramme in endlich vielen Variablen auf unendlich dimensionale Probleme verallgemeinert, wobei Zähler und Nenner sowie die Nebenbedingungsfunktionale als Integrale gegeben sind. Der Ansatz zur Dualität ist dadurch gekennzeichnet, daß das duale Problem wieder ein Quotientenprogramm ist und die gleiche Zielfunktion wie das primale Problem hat.

     

Duality theorems for a new class of multitime multiobjective variational problems

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (??, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.     

Polynomial algorithms for a class of linear programs

This paper describes several algorithms for solution of linear programs. The algorithms are polynomial when the problem data satisfy certain conditions.     

Duality for nonconvex absolute value programming and a characterization of linear max-min programs

In this paper a dual problem for nonconvex linear programs with absolute value functionals is constructed by means of a max-min problem involving bivalent variables. A relationship between the classical linear max-min problem and a linear program with absolute value functionals is developed. This program is then used to compute the duality gap between some max-min and min-max linear problems.     

Duality theorems for certain programs involving minimum or maximum operations

The relation of two mathematical programs—the so-called primal, (P) and dual, (D)—Are studied when (P) is a maximization of the minimum of two functions and (D) is the minimization of the maximum of two functions, and the function-pair of (D) are derived by certain, conjugate-like operators from that of (P). It is demonstrated that the weak duality holds when at least one of the pair is continuous. When further premises are met the strong duality is proved. From these results the usual Fenchel-duality may be deducted as well as few other primal-dual pairs.     

On a class of quadratic programs

This paper considers a class of quadratic programs where the constraints ae linear and the objective is a product of two linear functions. Assuming the two linear factors to be non-negative, maximization and minimization cases are considered. Each case is analyzed with the help of a bicriteria linear program obtained by replacing the quadratic objective with the two linear functions. Global minimum (maximum) is attained at an efficient extreme point (efficient point) of the feasible set in the solution space and corresponds to an efficient extreme point (efficient point) of the feasible set in the bicriteria space. Utilizing this fact and certain other properties, two finite algorithms, including validations are given for solving the respective problems. Each of these, essentially, consists of solving a sequence of linear programs. Finally, a method is provided for relaxing the non-negativity assumption on the two linear factors of the objective function.     

Strong duality for a special class of integer programs

A pair of primal-dual integer programs is constructed for a class of problems motivated by a generalization of the concept of greatest common divisor. The primal-dual formulation is based on a number-theoretic, rather than a Lagrangian, duality property; consequently, it avoids the dualitygap common to Lagrangian duals in integer programming.This research was partially supported by the National Science Foundation, Grant No. DCR-74-20584     

Optimality criterion for a class of nonlinear integer programs

Graver's optimality conditions based on Hilbert bases apply to an integer program with linear equations and a linear objective function. We generalize this result to include a fairly large class of nonlinear objective functions. Our extension provides in particular a link between the superadditivity of the difference-objective function and the Hilbert bases of conic subpartitions of .     

Branch-and-price for a class of nonconvex mixed-integer nonlinear programs

This work attempts to combine the strengths of two major technologies that have matured over the last three decades: global mixed-integer nonlinear optimization and branch-and-price. We consider a class of generally nonconvex mixed-integer nonlinear programs (MINLPs) with linear complicating constraints and integer linking variables. If the complicating constraints are removed, the problem becomes easy to solve, e.g. due to decomposable structure. Integrality of the linking variables allows us to apply a discretization approach to derive a Dantzig-Wolfe reformulation and solve the problem to global optimality using branch-andprice. It is a remarkably simple idea; but to our surprise, it has barely found any application in the literature. In this work, we show that many relevant problems directly fall or can be reformulated into this class of MINLPs. We present the branch-and-price algorithm and demonstrate its effectiveness (and sometimes ineffectiveness) in an extensive computational study considering multiple large-scale problems of practical relevance, showing that, in many cases, orders-of-magnitude reductions in solution time can be achieved.

     

A maximization method for a class of quasiconcave programs

An implementable linearized method of centers is presented for solving a class of quasiconcave programs of the form (P): maximizef ⁰(x), subject tox B andf ⁱ (x)0, for everyi{1, ...,m}, whereB is a convex polyhedral subset ofR ⁿ (Euclideann-space). Each problem function is a continuous quasiconcave function fromR ⁿ intoR ¹. Also, it is assumed that the feasible region is bounded and there existsx B such thatf _i (x) for everyi {1, ...,m}. For a broad class of continuous quasiconcave problem functions, which may be nonsmooth, it is shown that the method produces a sequence of feasible points whose limit points are optimal for Problem (P). For many programs, no line searches are required. Additionally, the method is equipped with a constraint dropping devise.The author wishes to thank a referee for suggesting the use of generalized gradients and a second referee whose detailed informative comments have enhanced the paper.This work was done while the author was in the Department of Mathematical Sciences at the University of Delaware.     

Duality for a class of fuzzy nonlinear optimization problem under generalized convexity

In this paper, a Mond-Weir type dual program for a nonlinear primal problem under fuzzy environment is formulated. The solution concept of primal-dual problems is inspired by the nondominated solution. We have considered ordering among fuzzy numbers as a partial ordering and using the concept of Hukuhara difference between two fuzzy numbers and $H$ -differentiability, appropriate duality theorems are established under pseudo/quasi-convexity assumptions. We have also illustrated a numerical example which satisfies the duality relations discussed in the paper.     

Duality and statistical tests of optimality for two stage stochastic programs

We present alternative methods for verifying the quality of a proposed solution to a two stage stochastic program with recourse. Our methods revolve around implications of a dual problem in which dual multipliers on the nonanticipativity constraints play a critical role. Using randomly sampled observations of the stochastic elements, we introduce notions of statistical dual feasibility and sampled error bounds. Additionally, we use the nonanticipativity multipliers to develop connections to reduced gradient methods. Finally, we propose a statistical test based on directional derivatives. We illustrate the applicability of these tests via some examples. This work was supported in part by Grant No. NSF-DMI-9414680 from the National Science Foundation