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The paper studies the second-order interval-valued variational problem under $$\eta $$ -bonvexity assumptions and proves the necessary optimality conditions. We investigate the functional which is $$\eta $$ -bonvex but not invex. Further, we prove the duality theorems i.e. the weak and strong duality theorem to relate the values of the primal problem and dual problem. To validate the credibility of the weak duality theorem, we formulate an example of a second-order interval-valued variational problem.

We aim to establish sufficient optimality conditions in terms of $$\text{ GA }$$ -stationary vectors and construct Wolfe and Mond–Weir types dual model in terms of contingent epiderivatives for the global optimal solution of nonsmooth semi-infinite mathematical programming problem with equilibrium constraints in finite-dimensional spaces ( $$\text{(NSIMPEC) }$$ for short). For this purpose, we provide some fundamental characterizations for the $$\varPsi $$ -preinvexity involving the notion of contingent epiderivative and contingent hypoderivative of extended-real-valued function and then some sufficient optimality conditions are obtained for the global optimal solution to such problem. For application purpose, a Mond–Weir and Wolfe types dual model for the problem $$\text{(NSIMPEC) }$$ are presented. Especially, some generalized Slater constraint qualifications are proposed and strong/weak duality theorems for the problem $$\text{(NSIMPEC) }$$ and its Mond–Weir and Wolfe types dual model are established. Some illustrative examples also proposed for our findings.

In this paper, we study the duality theorems of a nondifferentiable semi-infinite interval-valued optimization problem with vanishing constraints (IOPVC). By constructing the Wolfe and Mond–Weir type dual models, we give the weak duality, strong duality, converse duality, restricted converse duality, and strict converse duality theorems between IOPVC and its corresponding dual models under the assumptions of generalized convexity.

Duality theory is important in finding solutions to optimization problems. For example, in linear programming problems, the primal and dual problem pairs are closely related, i.e., if the optimal solution of one problem is known, then the optimal solution for the other problem can be obtained easily. In order for an optimization problem to be solved through the dual, the first step is to formulate its dual problem and analyze its characteristics. In this paper, we construct the dual model of an uncertain linear multi-objective optimization problem as well as its weak and strong duality criteria via conic duality. The multi-objective form of the problem is solved using the utility function method. In addition, the uncertainty is handled using robust optimization with ellipsoidal and polyhedral uncertainty sets. The robust counterpart formulation for the two uncertainty sets belongs to the conic optimization problem class; therefore, the dual problem can be built through conic duality. The results of the analysis show that the dual model obtained meets the weak duality, while the criteria for strong duality are identified based on the strict feasibility, boundedness, and solvability of the primal and dual problems.

Online interval scheduling problems consider scheduling a sequence of jobs on machines to maximize the total reward. Various approaches and algorithms have been proposed for different problem formulations. This paper provides a primal-dual approach to analyze algorithms for online interval scheduling problems. This primal-dual technique can be used for both stochastic and adversarial job sequences, and hence, is universally and generally applicable. We use strong duality and complementary slackness conditions to derive exact algorithms for scheduling stochastic equal-length job sequences on a single machine. We use weak duality to obtain upper bounds for the optimal reward for scheduling stochastic equal-length job sequences on multiple machines and C-benevolent job sequences on a single machine.

We examine a strong/weak duality between a Heisenberg coset of a theory with $\mathfrak{sl}_n$ subregular $\mathcal{W}$-algebra symmetry and a theory with a $\mathfrak{sl}_{n|1}$-structure. In a previous work, two of the current authors provided a path integral derivation of correlator correspondences for a series of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper, we derive correlator correspondences in a similar way but for a different series of generalized duality. This work is a part of the project to realize the duality of corner vertex operator algebras proposed by Gaiotto and Rap\v{c}\'ak and partly proven by Linshaw and one of us in terms of two dimensional conformal field theory. We also examine another type of duality involving an additional pair of fermions, which is a natural generalization of the fermionic FZZ-duality. The generalization should be important since a principal $\mathcal{W}$-superalgebra appears as its symmetry and the properties of the superalgebra are less understood than bosonic counterparts., Comment: 29 pages, final version to appear in JHEP

We consider a 4d scalar field coupled to large N free or critical O(N) vector models, either bosonic or fermionic, on a 3d boundary. We compute the β function of the classically marginal bulk/boundary interaction at the first non-trivial order in the large N expansion and exactly in the coupling. Starting with the free (critical) vector model at weak coupling, we find a fixed point at infinite coupling in which the boundary theory is the critical (free) vector model and the bulk decouples. We show that a strong/weak duality relates one description of the renormalization group flow to another one in which the free and the critical vector models are exchanged. We then consider the theory with an additional Maxwell field in the bulk, which also gives decoupling limits with gauged vector models on the boundary., SCOPUS: ar.j, info:eu-repo/semantics/published

The first part of this paper is devoted to the theory of Poisson-Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a duality pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak duality pairings between the tangent space and a subspace of the dual. We investigate related notion like Banach Lie bialgebras and Banach Poisson-Lie groups, suitably generalized to the non-reflexive Banach context. The second part of the paper is devoted to the treatment of particular examples of Banach Poisson-Lie groups related to the restricted Grassmannian and the KdV hierarchy. More precisely, we construct a Banach Poisson-Lie group structure on the unitary restricted Banach Lie group which acts transitively on the restricted Grassmannian. A "dual" Banach Lie group consisting of (a class of) upper triangular bounded operators admits also a Banach Poisson-Lie group structure of the same kind. We show that the restricted Grassmannian inherits a generalized Banach Poisson structure from the unitary Banach Lie group, called Bruhat-Poisson structure. Moreover the action of the triangular Banach Poisson-Lie group on it is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian.

This paper considers the mathematical programs with equilibrium constraints (MPEC). It is well-known that, due to the existence of equilibrium constraints, the Mangasarian-Fromovitz constraint qualification does not hold at any feasible point of MPEC and hence, in general, the developed numerical algorithms for standard nonlinear programming problems can not be applied to solve MPEC directly. During the past two decades, much research has been done to develop numerical algorithms and study optimality, stability, and sensitivity for MPEC. However, there are very few results on duality for MPEC in the literature. In this paper, we present a Wolfe-type duality for MPEC and, under some suitable conditions, we establish various duality theorems such as the weak duality, direct duality, converse duality, and strict converse duality theorems. We further show that a linear MPEC is equivalent to a linear programming problem in some sense.

The smash product V#H of a Hopf algebra H and an H-module vertex operator algebra V are investigated. A n ( V ) -theory and contragredient module theory are founded for V#H. If H is the group algebra of a finite subgroup of Aut ( V ) , all the irreducible inequivalent admissible V#H-modules are classified. We also prove the Maschke theorem in vertex operator algebra version. Finally, we prove weak duality theorems for actions and coactions in vertex operator algebras version.

In this article, we study the existence of Gf-bonvex/Gf -pseudo-bonvex functions and construct various nontrivial numerical examples for the existence of such type of functions. Furthermore, we formulate Mond-Weir type second-order nondifferentiable multiobjective programming problem and give a nontrivial concrete example which justify weak duality theorem present in the paper. Next, we prove appropriate duality relations under aforesaid assumptions.

The uncertainty for the continuous-time linear programming problem with time-dependent matrices is considered in this paper. In this case, the robust counterpart of the continuous-time linear programming problem is introduced. In order to solve the robust counterpart, it will be transformed into the conventional form of the continuous-time linear programming problem with time-dependent matrices. The discretization problem is formulated for the sake of numerically calculating the ϵ-optimal solutions, and a computational procedure is also designed to achieve this purpose.

This paper is devoted to the investigation of a nonsmooth multiobjective bilevel programming problem with equilibrium constraints ((MBPP) for short) in terms of convexificators in finite-dimensional spaces. We present necessary optimality conditions for the local weak efficient solution to such problem. Under the Mangasarian–Fromovitz and generalized standard Abadie type constraint qualification in the sense of convexificators, we establish as an application the Wolfe and Mond-Weir type dual problem for the problem (MBPP). Besides, we provide strong and weak duality theorems for the original problem and its Wolfe and Mond–Weir type dual problem under suitable assumptions on the $$\partial ^*$$ -convexity and the upper semi-regularity of objective and constraint functions. Illustrative examples are also proposed to demonstrate the main results of the paper.

An extended form of robust continuous-time linear programming problem with time-dependent matrices is formulated in this paper. This complicated problem is studied theoretically in this paper. We also design a computational procedure to solve this problem numerically. The desired data that appeared in the problem are considered to be uncertain quantities, which are treated according to the concept of robust optimization. A discretization problem of the robust counterpart is formulated and solved to obtain the ϵ-optimal solutions.

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.

Duality games are a way of looking at wave–particle duality. In these games. Alice and Bob together are playing against the house. The house specifies, at random, which of two sub-games Alice and Bob will play. One game, Ways, requires that they obtain path information about a particle going through an N-path interferometer and the other, Phases, requires that they obtain phase information. In general, because of wave–particle duality, Alice and Bob cannot always win the overall game. However, if the required amount of path and phase information is not too great, for example specifying a set of paths or phases, one of which is the right one, then they can always win. Here we study examples of duality games that can always be won, and develop a wave–particle duality relation expressed only in terms of mutual information to help analyze these games.

This article establishes a branch–bound technique to solve nonlinear convex–convex fractional multi-objective optimization problem in the non-convex feasible region. As far as the authors are concerned, this kind of problem is not solved by any other author in the literature. By transformation, multi-objective non-linear fractional problem is transformed into a multi-objective non-linear optimization problem. After giving preferences of weight to each objective, the original NLFMOOP is transformed into a nonlinear single-objective programming problem. Lagrange’s theorem of weak duality is used to find lower and upper bound for single objective nonlinear optimization problems in the feasible region. Some theoretical results for solving the multi-objective non-linear fractional problem have also been established. For showing the application of the proposed method, it has been applied to two numerical problems.

This paper is aimed at establishing a conjugate duality for the constrained optimizations equipped with some topical structures. First, we provide a dual problem for the general constrained optimization, discussing the weak duality as well as the strong duality based on the theory of abstract convexity. Transforming the zero duality gap property into a separation of two sets in image space, we involve the approach inspired by the image space analysis to study the dual theory by the aid of some separation functions. Then, using these results, we investigate this dual frame for the problem with some topical properties.

Many current deregulated markets allow multiple wind power producers (WPPs) to jointly offer energy in the short-term electricity market via an external agent such as a wind power aggregator. This paper proposes a budget balanced, fair, and stable framework to share the profit due to the joint wind energy offering, which is modeled as a core selection problem in the cooperative game theory. In particular, this design problem can be formulated as a large scale linear program with an exponential number of implicit constraints whose parameters are the outcomes of the wind coalition's optimal energy offering strategies. We propose a novel constraint generation algorithm to optimally solve this large scale optimization problem with affordable computation efforts. Different from the traditional computation methods, our proposed algorithm can effectively address the complexity involved in generating a feasible cut by exploiting the linear structure of the wind power offering model and the weak duality theory of linear programing. Extensive numerical results are then presented to illustrate the efficiency of the proposed framework in dealing with large scale aggregation of WPPs in a complex market framework and flexible adoption of optimization objectives.

Optimizing the sum-of-fractional functions under the bounded feasible space is a very difficult optimization problem in the research area of nonlinear optimization. All the existing solution methods in the literature are developed to find the solution of single-objective sum-of-fractional optimization problems only. Sum-of-fractional multi-objective optimization problem is not attempted to solve much by the researchers even when the fractional functions are linear. In the present article, a duality-based branch and bound computational algorithm is proposed to find a global efficient (non-dominated) solution for the sum-of-linear-fractional multi-objective optimization (SOLF-MOP) problem. Charnes–Cooper transformation technique is applied to convert the original problem into non-fractional optimization problem, and equivalence is shown between the original SOLF-MOP and non-fractional MOP. After that, weighted sum method is applied to transform MOP into a single-objective problem. The Lagrange weak duality theorem is used to develop the proposed algorithm. This algorithm is programmed in MATLAB (2016b), and three numerical illustrations are done for the systematic implementation. The non-dominance of obtained solutions is shown by comparison with the existing algorithm and by taking some feasible solution points from the feasible space in the neighborhood of obtained global efficient solution. This shows the superiority of the developed method.

In the present study, a new fuzzy based branch-bound approach is attempted for solving multi-objective linear fractional (MOLF) optimization problems. The original MOLF optimization problem is converted into equivalent fuzzy multi-objective linear fractional (FMOLF) optimization problem. Then branch and bound techniques is applied on FMOLF optimization problem. The feasible space of FMOLF optimization problem is bounded by triangular simplex space. The weak duality theorem is used to generate the bound for each partition and feasibility conditions are applied to neglect one of the partition in each step. After finite number of steps, a fuzzy efficient (Pareto-optimal) solution is obtained for FMOLF optimization problem which is also efficient (Pareto-optimal) solution of the original MOLF optimization problem. Some theoretical validations are also established for the proposed approach on FMOLF optimization problem. For the efficiency of proposed approach, it has been performed on two numerical applications. The method is coded in Matlab (2016). The results are compared with earlier reported methods.

In the present paper, we introduce a pair of second order fractional symmetric variational programs over cone constraints and derive weak, strong, and converse duality theorems under second order F-convexity assumptions. Moreover, self duality theorem is also discussed. Our results give natural unification and extension of some previously known results in the literature.

In this work, we study linear programming problems on time scales. This approach unifies discrete and continuous linear programming models and extends them to other cases ?in between?. After a brief introduction to time scales, we formulate the primal as well as the dual time scales linear programming models. Next, we establish and prove the weak duality theorem and the optimality conditions theorem for arbitrary time scales, while the strong duality theorem is established for isolated time scales. Finally, examples are given in order to illustrate the effectiveness of the presented results.

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 paper we give higher order sufficient optimality conditions for a fractional vector optimization problem over cones, using higher order cone-convex functions. A higher order Schaible type dual program is formulated over cones.Weak, strong and converse duality results are established by using the higher order cone convex and other related functions.

The notion of second order convexity and its generalization for $n$-set functions are introduced. Mond-weir type second order duality is formulated for the general mathematical programming problems involving $n$-set functions, and proved desired duality theorems. Further, counterexamples are illustrated in support of the present investigation.

We define a new class of optimal control problems and show that this class is the largest one of control problems where every admissible process that satisfies the Extended Pontryaguin Maximum Principle is an optimal solution of non-regular optimal control problems. In this class of problems the local and global minimum coincide. A dual problem is also proposed, which may be seen as a generalization of the Mond-Weir type dual problem, and it is shown that the 2-invexity notion is a necessary and sufficient condition in order to establish weak, strong and converse duality results between a non-regular optimal control problem and its dual problem. We also present an example to illustrate our results.

In this article, we have introduced a new class of higher-order $$(K\times Q)$$ -F-type I functions. Further, we have formulated two higher-order dual models, Wolfe and Schaible type, for a multiobjective fractional programming problem over arbitrary cones and proved appropriate duality relations under the higher-order $$(K\times Q)$$ -F-type I assumption. Nontrivial examples are also discussed to validate the weak duality results obtained in the 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.

Weak/strong duality is usually accompanied by what seems a puzzling ontological feature: the fact that under this kind of duality what is viewed as 'elementary' in one description gets mapped to what is viewed as 'composite' in the dual description. This paper investigates the meaning of this apparent 'particle democracy', as it has been called, by adopting an historical approach. The aim is to clarify the nature of the correspondence between 'dual particles' in the light of an historical analysis of the developments of the idea of weak/strong duality, starting with Dirac's electric-magnetic duality and its successive generalizations in the context of (Abelian and non-Abelian) field theory, to arrive at its first extension to string theory. This analysis is then used as evidential basis for discussing the 'elementary/composite' divide and, after taking another historical detour by analysing an instructive analogy case (DHS duality and related nuclear democracy), drawing some conclusions on the particle-democracy issue.

The resolution of integer programming problems is typically performed via branch and bound. Nodes of the branch-and-bound tree are pruned whenever the corresponding subproblem is proven not to contain a solution better than the best solution found so far. This is a key ingredient for achieving reasonable solution times. However, since subproblems are solved in floating-point arithmetic, numerical errors can occur and may lead to inappropriate pruning. As a consequence, optimal solutions may be cut off. We propose several methods for avoiding this issue, in the special case of a branch-cut-and-price formulation for the capacitated vehicle routing problem. The methods are based on constructing dual feasible solutions for the linear programming relaxations of the subproblems and obtaining, by weak duality, bounds on their objective function value. Such approaches have been proposed before for formulations with a small number of variables (dual constraints), but the problem becomes more complex when the number of variables is exponentially large, which is the case in consideration. We show that, in practice, along with being safe, our bounds are stronger than those usually employed, obtained with unsafe floating-point arithmetic plus some heuristic tolerance, and all of this at a negligible computational cost. We also discuss some potential advantages and other uses of our safe bounds derivation. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0747 .

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.

In this paper, the duality theory of a generalized quasi-equilibrium problem (also called generalized Ky Fan quasi-inequality) is investigated by using the image space approach. Generalized quasi-equilibrium problem is transformed into a minimization problem. The minimization problem is further reformulated as an image problem by virtue of linear/nonlinear separation function. The dual problem of the image problem is constructed in the image space, then zero duality gap between the image problem and its dual problem is derived under saddle point condition as well as the equivalent regular linear/nonlinear separation condition. Finally, some more sufficient conditions guaranteeing zero duality gap are also proposed.

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 introduce the concept of higher-order (Φ, ρ)-V-invexity and present two types of higher-order dual models for a semi-infinite minimax fractional programming problem. Weak, strong, and strict converse duality theorems are discussed under the assumptions of higher-order (Φ, ρ)-V-invexity to establish a relation between the primal and dual problems.

In this paper, we consider semi-infinite mathematical programming problems with equilibrium constraints (SIMPEC). We establish necessary and sufficient optimality conditions for the SIMPEC, using convexificators. We study the Wolfe type dual problem for the SIMPEC under the \(\partial ^{*}\)-convexity assumption. A Mond–Weir type dual problem is also formulated and studied for the SIMPEC under the \(\partial ^{*}\)-convexity, \(\partial ^{*}\)-pseudoconvexity and \(\partial ^{*}\)-quasiconvexity assumptions. Weak duality theorems are established to relate the SIMPEC and two dual programs in the framework of convexificators. Further, strong duality theorems are obtained under generalized standard Abadie constraint qualification.

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.

A new incremental transductive SVMs framework dependeding on duality is put forward for constrained optimization issues. Based on the weak duality theorem, the procedure of incremental learning is simplified for the task dual function increment. Based on this, two incremental learning methods are developed by updating limited dual parameters: (1) aggressive dual ascending; (2) local concave-convex procedure (LCCCP). Experiments demonstrated that our methods achieve comparable risk and accuracy to batch TSVMs, with less time consumption and memory requirment. Besides, our incremental learning methods can cope with concept drift and maintain smaller error rate than batch learning methods. The design and analysis of incremental semi-supervised learning methods is fully discussed in this research.

In this paper, we introduce the various types of generalized invexities, i.e., &alpha, f -invex/ &alpha, f -pseudoinvex and ( G , &alpha, f ) -bonvex/ ( G , &alpha, f ) -pseudobonvex functions. Furthermore, we construct nontrivial numerical examples of ( G , &alpha, f ) -bonvexity/ ( G , &alpha, f ) -pseudobonvexity, which is neither &alpha, f -bonvex/ &alpha, f -pseudobonvex nor &alpha, f -pseudoinvex with the same &eta, Further, we formulate a pair of second-order non-differentiable symmetric dual models and prove the duality relations under &alpha, f ) -pseudobonvex assumptions. Finally, we construct a nontrivial numerical example justifying the weak duality result presented in the paper.

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.

We have introduced higher order generalized hybrid B -(b,?,?,??,?r)-invex function. Then, we have estabilished higher order weak, strong and strict converse duality theorems for a multiobjective fractional programming problem with support function in the numerator of the objective function involving higher order generalized hybrid B -(b,?,?,??,?r)-invex functions. Our results extend and unify several results from the literature.

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...

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.

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.