Duality Results Research Articles

Revisiting the decomposition of cost efficiency for non-homothetic technologies: a directional distance function approach

In the early 1980’s Kopp and Diewert proposed a popular method to decompose cost efficiency into allocative and technical efficiency for parametric functional forms based on the radial approach initiated by Farrell. We show that, relying on recently proposed homogeneity and duality results, their approach is unnecessary for self-dual homothetic production functions, while it is inconsistent in the non-homothetic case. By stressing that for homothetic technologies the radial distance function can be correctly interpreted as a technical efficiency measure, since allocative efficiency is independent of the output level and radial input reductions leave it unchanged, we contend that for non-homothetic technologies this is not the case because optimal input demands depend on the output targeted by the firm, as does the inequality between marginal rates of substitution and market prices—allocative inefficiency. We demonstrate that a correct definition of technical efficiency corresponds to the directional distance function because its flexibility ensures that allocative efficiency is kept unchanged through movements in the input production possibility set when solving technical inefficiency, and therefore the associated cost reductions can be solely—and rightly—ascribed to technical-engineering-improvements. The new methodology allowing for a consistent decomposition of cost inefficiency is illustrated resorting to simple examples of non-homothetic production functions.

A set optimization approach to utility maximization under transaction costs

A set optimization approach to multi-utility maximization is presented, and duality results are obtained for discrete market models with proportional transaction costs. The novel approach allows us to obtain results for non-complete preferences, where the formulas derived closely resemble but generalize the scalar case.

Duality in convex minimum cost flow problems on infinite networks and hypernetworks

Minimum cost flow problems on infinite networks arise, for example, in infinite‐horizon sequential decision problems such as production planning. Strong duality for these problems was recently established for linear costs using an infinite‐dimensional Simplex algorithm. Here, we use a different approach to derive duality results for convex costs. We formulate the primal and dual problems in appropriately paired sequence spaces such that weak duality and complementary slackness can be established using finite‐dimensional proof techniques. We then prove, using a planning horizon proof technique, that the absence of a duality gap between carefully constructed finite‐dimensional truncations of the primal problem and their duals is preserved in the limit. We then establish that strong duality holds when optimal solutions to the finite‐dimensional duals are bounded. These theoretical results are illustrated via an infinite‐horizon shortest path problem. We also extend our results to infinite hypernetworks and apply this generalization to an infinite‐horizon stochastic shortest path problem. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(2), 98–115 2017

Optimal Skorokhod embedding given full marginals and Azéma–Yor peacocks

We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval $[0,1]$. The problem is related to the study of extremal martingales associated with a peacock (“process increasing in convex order,” by Hirsch, Profeta, Roynette and Yor [Peacocks and Associated Martingales, with Explicit Constructions (2011), Springer, Milan]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordère, Obłój, Spoida and Touzi [Ann. Appl. Probab. 26 (2016) 1–44]. Under technical conditions, we then characterize the optimal value and the solution to the dual problem. In particular, the optimal embedding corresponds to the Madan and Yor [Bernoulli 8 (2002) 509–536] peacock under their “increasing mean residual value” condition. We also discuss the associated martingale inequality.

Optimality and duality in multiobjective programming involving support functions

In this paper a vector optimization problem (VOP) is considered where each component of objective and constraint function involves a term containing support function of a compact convex set. Weak and strong Kuhn−Tucker necessary optimality conditions for the problem are obtained under suitable constraint qualifications. Necessary and sufficient conditions are proved for a critical point to be a weak efficient or an efficient solution of the problem (VOP) assuming that the functions belong to different classes of pseudoinvex functions. Two Mond Weir type dual problems are considered for (VOP) and duality results are established.

Generalized higher-order cone-convex functions and higher-order duality in vector optimization

In this paper, we introduce a new class of higher-order cone-convex, \((K_1, K_2)\)-pseudoconvex and quasiconvex functions which encapsulates several already known functions. Higher-order sufficient optimality conditions have been established for a vector optimization problem over cones by using these functions, under weaker conditions on multipliers as compared to other papers in this domain. Wolfe type and Mond–Weir type higher-order duals are formulated and corresponding duality results are established. A number of previously studied problems appear as special cases of our primal-dual models. In case of nonlinear programming problem, our higher-order duals reduce to the corresponding higher-order duals given by Mangasarian (J Math Anal Appl 51:607–620, 1975) and Mond and Zhang (Generalized convexity, generalized monotonicity: recent results. Kluwer, Dordrecht, pp 357–372, 1998).

OPTIMIZATION PROBLEMS WITH DIFFERENCE OF SET-VALUED MAPS UNDER GENERALIZED CONE CONVEXITY

In this paper, we establish the necessary and sufficient Karush-Kuhn-Tucker (KKT) conditions for an optimization problem with difference of set-valued maps under generalized cone convexity assumptions. We also study the duality results of Mond-Weir (MW D), Wolfe (W D) and mixed (Mix D) types for the weak solutions of the problem (P).

A twisted duality for parabolic Kazhdan–Lusztig R-polynomials

We prove a duality result for the parabolic Kazhdan–Lusztig R-polynomials of a finite Coxeter system. This duality is similar to, but different from, the one obtained in [9]. As a consequence of our duality we obtain an identity between the parabolic Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials of a finite Coxeter system. We also obtain applications to certain modules defined by Deodhar and derive a result that gives evidence in favor of Marietti's combinatorial invariance conjecture for parabolic Kazhdan–Lusztig polynomials.

Multi-objective semi-infinite variational problem and generalized invexity

In this paper, multiobjective semi-infinite variational problem (MSVP) has been considered. Relationship between efficiency, vector Kuhn Tucker point and vector Fritz John point for the stated problem have been established and authenticated by means of examples. Two duals namely Wolfe and Mond–Weir are purposed for (MSVP) and duality results are established under generalized invexity assumptions. Efficient solution of (MSVP) has been characterized by vector saddle point of a vector valued Lagrangian function of (MSVP). An equivalent (MSVP) is also constructed by a suitable modification of the objective function.

Duality for nondifferentiable minimax fractional programming problem involving higher order $$(\varvec{C},\varvec{\alpha}, \varvec{\rho}, \varvec{d})$$ ( C , α , ρ , d ) -convexity

In this paper, we present new class of higher-order \((C, \alpha , \rho , d)\)-convexity and formulate two types of higher-order duality for a nondifferentiable minimax fractional programming problem. Based on the higher-order \((C, \alpha , \rho , d)\)-convexity, we establish appropriate higher-order duality results. These results extend several known results to a wider class of programs.

Duality results for a second-order multiobjective fractional programming problem with generalised convexity

The purpose of this paper is to study a class of non-differentiable multiobjective fractional programming problem in which every component of objective functions contains a term involving the support function of a compact convex set. For a differentiable function, we use the definition of second-order (C, α, ρ, d) - V-type-I convex function. Further, a Mond-Weir type dual has been formulated for this problem and appropriate duality results have been proved under second-order (C, α, ρ, d) - V-type-I convexity assumptions.

Towards generalized cohmology Schubert calculus via formal root polynomials

An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. In this paper we define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We focus on the case of the hyperbolic formal group law (corresponding to elliptic cohomology). We study some of the properties of formal root polynomials. We give applications to the efficient computation of the transition matrix between two natural bases of the formal Demazure algebra in the hyperbolic case. As a corollary, we rederive in a transparent and uniform manner the formulas of Billey and Graham-Willems. We also prove the corresponding formula in connective $K$-theory, which seems new, and a duality result in this case. Other applications, including some related to the computation of Bott-Samelson classes in elliptic cohomology, are also discussed.

Necessary and Sufficient Conditions for Weak Quasi Efficiency in Nonsmooth Multiobjective Problems

ABSTRACTIn this article, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given and the relations between them are analyzed. We establish Kuhn-Tucker and strong Kuhn-Tucker necessary optimality conditions for (weak) quasi efficiency in terms of the Clarke subdifferential. By using two new classes of generalized convex functions, sufficient conditions for local (weak) quasi efficient are also provided. Furthermore, we study the Mond-Weir type dual problem and establish weak, strong and converse duality results.

Characterization of Generalized Invexity in Multiobjective Fractional Variational Problem

In this article we define certain conditions on the functionals of multi-objective fractional variational problem in order that it becomes F-Kuhn Tucker pseudo invex or F-Fritz John pseudo invex. We also define F-KT and F-FJ points. Further, these problems are characterized such that all F-KT and F-FJ points become efficient solutions for the featured problem. An example is presented to verify the existence of F-KT point. A Parametric dual is proposed and various duality results are proved under the assumption of F-KT as well as F-FJ pseudo invexity.

Sufficiency and mixed type duality for multiobjective variational control problems involving <i>α</i>-V-univexity

In this paper, we focus our study on a multiobjective variational control problem and establish sufficient optimality conditions under the assumptions of α -V-univex function. Furthermore, mixed type duality results are also discussed under the aforesaid assumption in order to relate the primal and dual problems. Examples are given to show the existence of α -V-univex function and to elucidate duality result.

Approximate quasi efficiency of set-valued optimization problems via weak subdifferential

This paper deals with the approximate quasi efficient solutions of set-valued optimization problems. We establish the necessary and sufficient optimality conditions for \(\epsilon e\)-quasi efficiency via weak subdifferential under some approximate cone convexity assumptions. We also study duality results of Wolfe and Mond-Weir types for Geoffrion \(\epsilon e\)-quasi efficiency of finite dimensional set-valued optimization problems.

On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems

In this paper, we consider a semi-infinite multiobjective optimization problem with more than two differentiable objective functions and uncertain constraint functions, which is called a robust semi-infinite multiobjective optimization problem and give its robust counterpart \({\mathrm{(RSIMP)}}\) of the problem, which is regarded as the worst case of the uncertain semi-infinite multiobjective optimization problem. We prove a necessary optimality theorem for a weakly robust efficient solution of \({\mathrm{(RSIMP)}} \), and then give a sufficient optimality theorem for a weakly robust efficient solution of \({\mathrm{(RSIMP)}}\). We formulate a Wolfe type dual problem of \({\mathrm{(RSIMP)}}\) and give duality results which hold between \({\mathrm{(RSIMP)}}\) and its dual problem.

Pseudoproduction, Pseudocost and Profit Functions in Monopoly from the Dual Perspective

Abstract Background: Duality in the microeconomic theory enables us to represent consumers’ preferences and production technology with various dual functions satisfying certain regularity conditions. Objectives: Since the basis for the application of duality in the microeconomic theory is the price taking behaviour, this paper takes the challenge of applying principles of duality to a monopolistic case where a single producer has an influence on the price which it charges for its product. Methods/Approach: The standard approach of deriving the profit function for the monopolist from the production function and the defined pseudoproduction function is accompanied by an alternative approach in which the starting point is the pseudocost function. Starting from the derived profit function, the pseudoproduction function and the pseudocost functions are recovered and a version of Hotelling’s lemma is given. Results: The structure of the profit maximization problem in a monopolistic case was made similar to the structure of the profit maximization problem in the perfectly competitive case and it is shown that all starting functions can be recovered back from derived functions. A version of Hotelling’s lemma is illustrated, which brings us indirectly from the profit function to the supply function. Conclusions: By introducing the pseudoproduction function in the profit maximization model of a monopolist, the structure of the problem becomes similar to the perfectly competitive case and duality results can be applied. The profit function is derived from the pseudoproduction and the pseudocost function, and all starting functions are recovered back from the derived profit function.

Second-order multiobjective programming problems and symmetric duality relations with $${G_{f}}$$ G f -bonvexity

In this paper, we have introduced the definition of \({G_{f}}\)-bonvex function. To show the existence of such a function, we have also discussed a non-trivial example. Further, second-order G-Wolfe and G-Mond–Weir type primal-dual pairs have been formulated and appropriate duality results have been proved under aforesaid assumptions.

Wholesale Most-Favored-Nation Clauses and Third Degree Price Discrimination with Negative Consumption Externalities: A Duality Result

Most-favored-nation (MFN) clauses in wholesale contracts have been the subject of recent controversy and renewed antitrust scrutiny. We demonstrate that an environment where MFN clauses may be widely adopted shares a duality property with an environment where a monopolist sells a good exhibiting negative consumption externalities directly to consumers. Many arguments for or against wholesale MFN clauses can be re-framed in this consumer framework where the welfare effects closely follow those developed in the third degree price discrimination literature by Schmalensee (1981) and Varian (1985). Specifically, the risk that MFN clauses deter entry in the producer problem can be inter- preted equivalently in the consumer problem as the risk that a ban on price discrimination results in low types no longer being served, and also reduces welfare. We argue that ex- amining wholesale MFNs from this perspective offers a greatly simplified approach: the welfare effects of wholesale MFNs are in general ambiguous for the same reason the effect of uniform pricing on welfare is ambiguous in Schmalensee (1981) and Varian (1985).