Strong Duality Results Research Articles

Symmetric duality with invexity in variational problems

Weak and strong duality results are established under invexity hypotheses for symmetric dual variational problems without positivity constraints. Self-dual problems and static symmetric dual programs are included as special cases.

Multiobjective Programming Strong Vector Duality

Abstract The notion of convex programming with set inclusive constraints and objective set is extended and generalized from a linear, scalar valued case to nonlinear multiobjective programming with nonlinear constraints No differentiability or subdifferentiability is required. Any form of convexity is not needed either. Some strong vector duality results are established without any scalarization.

Generalized invexity for nonsmooth vector-valued mappings

We define four types of invexity for Lipschitz vector-valued mappings from Rp to Rq that generalize previous definitions of invexity in the differentiable setting. After establishing relationships between the various definitions, we show the importance of the concept of nonsmooth invexity in the field of optimization. In particular, we obtain conditions sufficient for optimality in unconstrained and cone-constrained nondifferentiable programming that are weaker than previous conditions presented in the literature; we also obtain weak and strong duality results.

Duality Theory In Multiobjective Differentiable Programming

Abstract In this paper, a Mond-Weir type dual is considered for a multiobjective n n-linear programming problem with inequality and equality constraints. Both the primal and the dual have the same objective function. Weak and strong duality theorems are established along the lines of classical nonlinear programming. This is done under various (relaxed) convexity restrictions on the linear combinations of the components of maps describing the constraints and the objective function. Some of the strong duality results are based on necessary conditions for a vector to be a Pareto minimal. These necessary conditions are based on the idea of a convergence vector, a constraint qualification and a theorem of the alternative.

Stationary dual prices and depreciation

This paper models a machine replacement and capacity expansion problem as an infinite-horizon linear program. We establish a strong duality result and show that stationary dual prices are optimal, regardless of initial conditions. These prices measure the economic value of owning vintage machinery and thus define depreciation schedules. We present necessary and sufficient conditions for straight-line depreciation.

The Round-Trip p-Center and Covering Problem on a Tree

A task performed by a vehicle consists of picking up goods at a given client and delivering it to another client. The round-trip distance is the distance traveled by the vehicle between the departure from and the return to its depot. The round-trip p-center problem is to locate p depots with respect to a finite number of pairs of clients so as to minimize the maximum round-trip cost, which is defined as a nonlinear increasing function of the round-trip distance from the nearest depot. The round-trip covering problem is to find the minimum number of depots such that each round-trip cost is no more than a given constant. For both problems a dual problem is defined and in case the underlying network is a tree a strong duality result is proved. We give algorithms to solve both tree round-trip location problems in polynomial time.

On abstract duality in mathematical programming

It is shown that duality in mathematical programming can be treated as a purely order theoretic concept which leads to some applications in economics. Conditions for strong duality results are given. Furthermore the underlying sets are endowed with (semi-)linear structures, and the perturbation function of arising linear and integer problems, which include bottleneck problems and extremal problems (in the sense of K. Zimmermann), is investigated.

Duality for Algebraic Linear Programming

A duality theory for algebraic linear (integer) programming (ALP) is developed which is of the same importance for linear (integer) programming with linear algebraic objectives as linear programming duality is for classical LP. In particular, optimality criteria for primal, primal-dual, and dual methods are given which generalize feasibility and complementarity criteria of classical LP. Strong duality results are given for special combinatorial problems. Further, the validity and finiteness of a primal simplex method based on a feasibility criterion are proved in the case of nondiscrete variables. In this case a strong duality result is shown.