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Altres ajuts: Australian Research Council DP140103213 In this work, we achieve a complete characterization of the existence of a saddle value, for bifunctions which are convex, proper, and lower semi continuous in their first argument, by considering new suitably defined notions of special directions of recession. As special cases, we obtain some recent results of Lagrangian duality theory on zero duality gap for convex programs.

Conservative dynamical systems propagate as stationary points of the action functional. Using this representation, it has previously been demonstrated that one may obtain fundamental solutions for two-point boundary value problems for some classes of conservative systems via solution of an associated dynamic program. Further, such a fundamental solution may be represented as a set of solutions of differential Riccati equations (DREs), where the solutions may need to be propagated past escape times. Notions of “static duality” and “stat-quad duality” are developed, where the relationship between the two is loosely analogous to that between convex and semiconvex duality. Static duality is useful for smooth functionals where one may not be guaranteed of convexity or concavity. Some simple properties of this duality are examined, particularly commutativity. Application to stationary action is considered, which leads to propagation of DREs past escape times via propagation of stat-quad dual DREs.

Following the idea of the conjecture for semi-infinite programming in a paper by Kortanek and Zhang, recently published in Optimization, in this paper we show that the Fourier–Motzkin elimination is not needed in the study of the strong duality and dual pricing properties for semi-infinite programming. We also prove several new results on the strong duality and dual pricing properties. Specifically, we propose a new subspace, under which the strong duality property holds. We give a necessary and sufficient condition for the dual pricing property to hold under this subspace, which is further used to examine the examples presented in the Basu–Martin–Ryan paper.

In this note, we use recent zero duality results arising from Monotropic Programming problem for analyzing consistency of the convex feasibility problem in Hilbert spaces. We characterize consistency in terms of the lower semicontinuity of the infimal convolution of the associated support functions.

In the framework of conjugate duality we discuss nonlinear and linear single minimax location problems with geometric constraints, where the gauges are defined by convex sets of a Frechet space. The version of the nonlinear location problem is additionally considered with set-up costs. Associated dual problems for this kind of location problems will be formulated as well as corresponding duality statements. As conclusion of this paper, we give a geometrical interpretation of the optimal solutions of the dual problem of an unconstraint linear single minimax location problem when the gauges are a norm. For an illustration, an example in the Euclidean space will follow.

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, 2017

Primal or dual strong-duality (or min-sup, inf-max duality) in nonconvex optimization is revisited in view of recent literature on the subject, establishing, in particular, new characterizations for the second case. This gives rise to a new class of quasiconvex problems having zero duality gap or closedness of images of vector mappings associated to those problems. Such conditions are described for the classes of linear fractional functions and that of quadratic ones. In addition, some applications to nonconvex quadratic optimization problems under a single inequality or equality constraint, are presented, providing new results for the fulfillment of zero duality gap or dual strong-duality.

In this paper, by virtue of the epigraph technique, we first introduce some new regularity conditions and then obtain some complete characterizations of the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for a new class of DC optimization involving a composite function. Moreover, we apply the strong and stable strong duality results to obtain some extended (stable) Farkas lemmas and (stable) alternative type theorems for this DC optimization problem. As applications, we obtain the corresponding results for a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator, respectively.

The paper deals with the minimization of a function over the solution set of a range inclusion problem determined by a multifunction. A strong Lagrange duality is provided first in terms of a quasirelative interior condition and then under a so-called Assumption (S). When the function and the multifunction are convex, we improve this duality under a closed cone condition. The stability analysis is investigated. In addition, if the multifunction is a convex process, then the Fenchel dual is performed in terms of its conjugate. As a first application, we provide a unified approach to the optimization of general discrete inclusions systems; in particular, we improve several results on optimal control, strong Lagrange duality and Fenchel duality for some classes of convex controlled discrete processes.

The aim of this paper is to establish a fundamental theory of convex analysis for the sets and functions over a discrete domain. By introducing conjugate/biconjugate functions and a discrete duality notion for the cones over discrete domains, we study duals of optimization problems whose decision parameters are integers. In particular, we construct duality theory for integer linear programming, provide a discrete version of Slater’s condition that implies the strong duality and discuss the relationship between integrality and discrete convexity.

In this paper, we consider multiobjective semi-infinite optimization problems which are defined in a finite-dimensional space by finitely many objective functions and infinitely many inequality constraints. We present duality results both for the convex and nonconvex case. In particular, we show weak, strong and converse duality with respect to both efficiency and weak efficiency. Moreover, the property of being a locally properly efficient point plays a crucial role in the nonconvex case.

In this paper, we consider the extended Celis--Dennis--Tapia (CDT) problem that has a positive duality gap. It is presented in theory that this positive duality gap can be narrowed by adding an app...

This paper deals with the Quasi Variational Inequality (QVI) problem on Banach spaces. Necessary and sufficient conditions for the solutions of QVI are given, using the subdifferential of distance function and the normal cone. A dual problem corresponding to QVI is constructed and strong duality is established. The solutions of dual problem are characterized according to the saddle points of the Lagrangian map. A gap function for dual of QVI is presented and its properties are established. Moreover, some applied examples are addressed.

In this paper we show a duality theorem for a b -complementary multisemigroup master problem.

In this article, we present a conjugate duality for nonconvex optimization problems. This duality scheme is symmetric and has zero gap. As applied to a vector-maximization problem, it transforms the latter into an optimization problem over a weakly efficient set which can be solved by monotonic optimization methods.

In this paper, we consider an optimization problem with geometric and cone constraints, whose objective function is a composition of $$n+1$$ functions. For this problem, we calculate its conjugate dual problem, where the functions involved in the objective function of the primal problem will be decomposed. Furthermore, we formulate generalized interior point regularity conditions for strong duality and give necessary and sufficient optimality conditions. As applications of this approach, we determine the formulas of the conjugate as well as the biconjugate of the objective function of the primal problem and discuss an optimization problem having as objective function the sum of reciprocals of concave functions.

Third order dual of a primal nonlinear programming problem is established which involves the third order derivatives of the functions constituting the primal problem. Desired duality theorems are provided for the pair of primal and the corresponding third order dual problem. Numerical examples are illustrated to justify the efficiency of the proposed method. It is also observed that some of the existing results are obtained as special cases.

The concept of duality for the variational problems is introduced in general Banach spaces. Different forms of duality such as Mangasarian and Mond-Weir type are studied. Mond-Weir type duality is considered to weaken the convexity requirements. Many duality (weak, strong and converse) results are established under generalized convexity assumptions. Again, examples and counterexamples are discussed in support of the investigation.

In this paper, we introduce the notion of a second-order cone- convex function involving second-order directional derivative. Also, second-order cone-pseudoconvex, second-order cone-quasiconvex and other related functions are defined. Second-order optimality and Mond-Weir type duality results are derived for a vector optimization problem over conesnusing the introduced classes of functions.

This paper is devoted to the study of optimality conditions and duality theory for a set-valued optimization problem. by using the higher-order radial derivative of a set-valued map, we establish Fritz John and Kuhn-Tucker types necessary and sufficient optimality conditions for a weak minimizer of a set-valued optimization problem under the assumption that set-valued maps in the formulation of objective and constraint maps are near cone-subconvexlike. As an application of the optimality conditions, we prove weak, strong and converse duality theorems for Mond-Weir and Wolfe types dual problems.

The concept of duality gap function in infinite dimensional linear programming is considered in this paper. Basic properties of the function and two theorems on its behavior are obtained by using duality theorems with interior conditions. As illustrations for the results, we investigate the parametric versions of an example due to D. Gale and parametric linear programs on spaces of continuous functions. The notions of Riemann–Stieltjes integral and function of bounded variation have been shown to be very useful for our investigations.

In this paper, a pair of Mond-Weir type higher-order symmetric dual programs over arbitrary cones is formulated. The appropriate duality theorems, such as weak duality theorem, strong duality theorem and converse duality theorem, are established under higher-order (strongly) cone pseudoinvexity assumptions.

We investigate the augmented Lagrangian dual (ALD) for mixed integer linear programming (MIP) problems. ALD modifies the classical Lagrangian dual by appending a nonlinear penalty function on the violation of the dualized constraints in order to reduce the duality gap. We first provide a primal characterization for ALD for MIPs and prove that ALD is able to asymptotically achieve zero duality gap when the weight on the penalty function is allowed to go to infinity. This provides an alternative characterization and proof of a recent result in Boland and Eberhard (Math Program 150(2):491---509, 2015, Proposition 3). We further show that, under some mild conditions, ALD using any norm as the augmenting function is able to close the duality gap of an MIP with a finite penalty coefficient. This generalizes the result in Boland and Eberhard (2015, Corollary 1) from pure integer programming problems with bounded feasible region to general MIPs. We also present an example where ALD with a quadratic augmenting function is not able to close the duality gap for any finite penalty coefficient.

The advantage of second-order duality is that if a feasible point of the primal is given and first-order duality conditions are not applicable (infeasible), then we may use second-order duality to provide a lower bound for the value of primal problem. Consequently, it is quite interesting to discuss the duality results for the case of second order. Thus, we focus our study on a discussion of duality relationships of a minimax fractional programming problem under the assumptions of second order B-(p, r)-invexity. Weak, strong and strict converse duality theorems are established in order to relate the primal and dual problems under the assumptions. An example of a non trivial function has been given to show the existence of second order B-(p, r)-invex functions.

We present a new duality theory for non-convex variational problems, under possibly mixed Dirichlet and Neumann boundary conditions. The dual problem reads nicely as a linear programming problem, and our main result states that there is no duality gap. Further, we provide necessary and sufficient optimality conditions, and we show that our duality principle can be reformulated as a min–max result which is quite useful for numerical implementations. As an example, we illustrate the application of our method to a celebrated free boundary problem. The results were announced in Bouchitte and Fragala (C R Math Acad Sci Paris 353(4):375–379, 2015).

Duality theory is pervasive in finite-dimensional optimization. There is growing interest in solving infinite-dimensional optimization problems and hence a corresponding interest in duality theory in infinite dimensions. Unfortunately, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. In finite dimensions, a dual solution is represented by a vector of “dual prices” that index the primal constraints and have a natural economic interpretation. In infinite dimensions, we show that this simple dual structure, and its associated economic interpretation, may fail to hold for a broad class of problems with constraint vector spaces that are Riesz spaces (ordered vector spaces with a lattice structure) that either are $\sigma$-order complete or satisfy the projection property. In these spaces we show that the existence of interior points required by common constraint qualifications for zero duality gap (such as Slater's condition) implies the existence of ...

For a general class of non-convex optimization problems, a class of power reformulation closes the duality gap between the primal problem and its Lagrangian dual, when the order of the power is sufficiently large. In this paper, we first estimate a lower bound of the power above which the attainment of the zero duality gap can be ensured. After introducing a suitable shifting, we further show, surprisingly, that order three is always sufficient to guarantee the zero duality gap. We then extend the proposed shifted power reformulation to discrete optimization.

In this paper, we establish a strong duality theorem for a Mond-Weir type multiobjective higher order nondifferentiable symmetric dual programs. Our work relaxes the hypotheses used to prove the strong duality result (by omitting one of the condition (hypothesis (IV)), Theorem 2.1) in the recent paper (Yang et al. J. Ind. Manag. Optim. 9, 525–530, (2013)).

In this paper, we consider a generalized fractional robust optimization problem (FP). Establishing a nonfractional optimization problem (NFP) equivalent to (FP), we establish necessary optimality conditions and duality results.

In this article, a variational control problem is considered. We introduce the concept of higher order (F,G,ρ)-convex function for a continuous case. Further, we formulate higher order mixed-type dual for the considered problem and obtain appropriate duality results using aforesaid assumptions.

The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

In this work, we study the class of problems called semi-continuous optimization, which contains constrained minimization (maximization) problems with lower (upper) semi-continuous objective functions. We show some existence conditions for solutions based on asymptotic techniques, as well as a duality scheme based on the Fenchel---Moreau conjugation specifically applied to semi-continuous problems. Promising results are obtained, when we apply this scheme to minimize quadratic functions (whose Hessians can be symmetric indefinite) over nonempty, closed and convex polyhedral sets.

In the present article, we formulate two different kinds of higher-order dual models related to the multi-objective programming problem containing arbitrary norms. Furthermore, weak, strong and strict converse duality results are established under the assumptions of higher-order ( Φ , ρ ) -invex function. Results obtained in this paper unify and extend some previously known results in the literature.

We consider classical problems of the calculus of variations of the kind (1) I ( Ω ) : = inf ⁡ { ∫ Ω f ( u , ∇ u ) d x + ∫ Γ 1 γ ( u ) d H N − 1 , u = u 0 on Γ 0 } where Ω is an open bounded subset of R N , ( Γ 0 , Γ 1 ) is a partition of ∂Ω, γ is a Lipschitz function, and f = f ( t , z ) is an l.s.c. function satisfying suitable growth conditions, which is convex in z , but possibly not in t . We present a new duality theory in which the dual problem reads quite nicely as a linear programming problem. The solvability of such a dual problem is a major issue. It can be achieved in the one-dimensional case, and in higher dimensions under special assumptions on f . Our results apply to phase transition and free-boundary problems.

We introduce a notion of $k$th order stochastic monotonicity and duality that allows one to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put - call symmetries in option pricing. Our general $k$th order duality can be financially interpreted as put - call symmetry for powered options. The main objective of the present paper is to develop an effective analytic approach to the analysis of duality leading to the full characterization of $k$th order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put -call symmetries., Comment: To appear in Journal of Applied Probability 52:1 (March 2015)

Recently, a duality web for three dimensional theories with Chern-Simons terms was proposed. This can be derived from a single bosonization type duality, for which various supporting arguments (but not a proof) were given. Here we explicitly derive this bosonization, in the Abelian case and for a particular regime of parameters. To do this, we use the particle-vortex duality in combination with a Buscher-like duality (both considered in the regime of low energies). As a corollary, Son's conjectured duality is derived in a somewhat singular limit of vanishing mass., 10 pages, Latex

This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones. This research was supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam, Project 101.01-2015.27, Generalizations of Farkas lemma with applications to optimization, by the Ministry of Economy and Competitiveness of Spain and the European Regional Development Fund (ERDF) of the European Commission, ProjectMTM2014-59179-C2-1-P, and by the Australian Research Council, Project DP160100854.

In this paper we consider the duality gap function g that measures the difference between the optimal values of the primal problem and of the dual problem in linear programming and in linear semi-infinite programming. We analyze its behavior when the data defining these problems may be perturbed, considering seven different scenarios. In particular we find some stability results by proving that, under mild conditions, either the duality gap of the perturbed problems is zero or + ∞ around the given data, or g has an infinite jump at it. We also give conditions guaranteeing that those data providing a finite duality gap are limits of sequences of data providing zero duality gap for sufficiently small perturbations, which is a generic result. This research was partially supported by MINECO of Spain and FEDER of EU, Grant MTM2014-59179-C2-01 and SECTyP-UNCuyo Res. 4540/13-R.

The aim of this paper is to introduce the concept of second-order (F, α, ρ, θ)-convexity for variational problems. An example is constructed to examine the existence of the introduced class of functions, which is more general than previously defined functions. In order to relate the primal variational problem and its second order dual, several duality results, viz., weak, strong and strict converse duality results are established.

In the present paper, we examine duality results for Wolfe-type second-order fractional symmetric dual programs. These duality results are then used to investigate minimax mixed integer symmetric dual fractional programs. We also discuss self-duality results at the end.

A generalized dual to a control problem containing support functions is formulated and various duality theorems are established under generalized convexity hypotheses. This dual model represents the combination of Wolfe and Mond-Weir type dual models to the control problem and hence it is described as a generalized dual. Some special cases are obtained. A close relationship of duality results with those of the nonlinear programming problems involving support functions is indicated.

This paper addresses the duality theory of a nonlinear optimization model whose objective function and constraints are interval valued functions. Sufficient optimality conditions are obtained for the existence of an efficient solution. Three type dual problems are introduced. Relations between the primal and different dual problems are derived. These theoretical developments are illustrated through numerical example.

A class of second order ( F , α , ρ , d , E )-convex functions and their generalization on functions involved, weak, strong, and converse duality theorems are established for a second order Mond-Weir type dual problem.

This paper deals with multi-commodity flow problem with fractional objective function. The optimality conditions and the duality concepts of this problem are given. For this aim, the fractional linear programming formulation of this problem is considered and the weak duality, the strong direct duality and the weak complementary slackness theorems are proved applying the traditional duality theory of linear programming problems which is different from same results in Chadha and Chadha (2007) [1]. In addition, a strong (strict) complementary slackness theorem is derived which is firstly presented based on the best of our knowledge. These theorems are transformed in order to find the new reduced costs for fractional multi-commodity flow problem. These parameters can be used to construct some algorithms for considered multi-commodity flow problem in a direct manner. Throughout the paper, the boundedness of the primal feasible set is reduced to a weaker assumption about solvability of primal problem which is another contribution of this paper. Finally, a real world application of the fractional multi-commodity flow problem is presented.

KHUSHBOO VERMA AND T. R. GULATIAbstract. In this paper, we introduce a pair of higher-order symmetricdual models/problems. Weak, strong and converse duality theorems forthis pair are established under the assumption of higher-order invexity.Moreover, self duality theorem is also discussed.AMS Mathematics Subject Classi cation : 90C46, 49N15, 90C30.

Mond-Weir type duality for control problem with support functions is investigated under generalized convexity conditions. Special cases are derived. A relationship between our results and those of nonlinear programming problem containing support functions is outlined.