Multiobjective duality for convex ratios

In this paper we present a duality approach for a multiobjective fractional programming problem. The components of the vector objective function are particular ratios involving the square of a convex function and a positive concave function. Applying the Fenchel-Rockafellar duality theory for a scalar optimization problem associated to the multiobjective primal, a dual problem is derived. This scalar dual problem is formulated in terms of conjugate functions and its structure gives an idea about how to construct a multiobjective dual problem in a natural way. Weak and strong duality assertions are presented.     

New strong duality results for convex programs with separable constraints

It is known that convex programming problems with separable inequality constraints do not have duality gaps. However, strong duality may fail for these programs because the dual programs may not attain their maximum. In this paper, we establish conditions characterizing strong duality for convex programs with separable constraints. We also obtain a sub-differential formula characterizing strong duality for convex programs with separable constraints whenever the primal problems attain their minimum. Examples are given to illustrate our results.     

Farkas-type results for inequality systems with composed convex functions via conjugate duality

We present some Farkas-type results for inequality systems involving finitely many functions. Therefore we use a conjugate duality approach applied to an optimization problem with a composed convex objective function and convex inequality constraints. Some recently obtained results are rediscovered as special cases of our main result.     

Wolfe duality and Mond-Weir duality via perturbations

Considering a general optimization problem, we attach to it by means of perturbation theory two dual problems having in the constraints a subdifferential inclusion relation. When the primal problem and the perturbation function are particularized different new dual problems are obtained. In the special case of a constrained optimization problem, the classical Wolfe and Mond-Weir duals, respectively, follow as particularizations of the general duals by using the Lagrange perturbation. Examples to show the differences between the new duals are given and a gate towards other generalized convexities is opened.     

Generalized concavity and duality with a square root term

Duality results for a class of nondifferentiable mathematical programming problems are given. These results allow for the weakening of the usual convexity conditions required for duality to hold. A pair of symmetric and self dual nondifferentiable programs under weaker convexity conditions are also given. A subgradient symmetric duality is proposed and its limitations discussed. Finally, a pair of nondifferentiable mathematical programs containing arbitrary norms is presented.     

On generalized convexity and duality with a square root term

We extend the duality theorems for a class of nondifferentiable problems with Mond-Weir type duals.     

On strong and total Lagrange duality for convex optimization problems

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.     

Stable zero duality gaps in convex programming: Complete dual characterisations with applications to semidefinite programs

The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite-dimensional problems and it frequently fails for problems with non-polyhedral constraints such as the ones in semidefinite programming problems. Over the years, various criteria have been developed ensuring zero duality gaps for convex programming problems. In the present work, we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem. Globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions. For convex semidefinite programs, we establish necessary and sufficient dual conditions for stable zero duality gaps, as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right-hand side and convex objective function. Zero duality gap results for second-order cone programming problems are also given. Our approach makes use of elegant conjugate analysis and Fenchel's duality.     

On the relations between different duals assigned to composed optimization problems

For an optimization problem with a composed objective function and composed constraint functions we determine, by means of the conjugacy approach based on the perturbation theory, some dual problems to it. The relations between the optimal objective values of these duals are studied. Moreover, sufficient conditions are given in order to achieve equality between the optimal objective values of the duals and strong duality between the primal and the dual problems, respectively. Finally, some special cases of this problem are presented.      

On duality theory of convex semi-infinite programming

In this article we discuss weak and strong duality properties of convex semi-infinite programming problems. We use a unified framework by writing the corresponding constraints in a form of cone inclusions. The consequent analysis is based on the conjugate duality approach of embedding the problem into a parametric family of problems parameterized by a finite-dimensional vector.     

A strong duality theorem for the minimum of a family of convex programs

We produce a duality theorem for the minimum of an arbitrary family of convex programs. This duality theorem provides a single concave dual maximization and generalizes recent work in linear disjunctive programming. Homogeneous and symmetric formulations are studied in some detail, and a number of convex and nonconvex applications are given.This work was partially funded by National Research Council of Canada, Grant No. A4493. Thanks are due to Mr. B. Toulany for many conversations and to Dr. L. MacLean who suggested the chance-constrained model.     

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments     

The gap function of a convex program

The gap function expresses the duality gap of a convex program as a function of the primal variables only. Differentiability and convexity properties are derived, and a convergent minimization algorithm is given. An example gives a simple one-variable interpretation of weak and strong duality. Application to user-equilibrium traffic assignment yields an appealing alternative optimization problem.     

On the Kato square root problem

In the infinite-dimensional separable complex Hilbert space we construct new abstract examples of unbounded maximal accretive and maximal sectorial operators B for which $\mathrm{dom}{B}^{\frac{1}{2}}\ne \mathrm{dom}{B}^{?\frac{1}{2}}$. New criterions for the equality are established.     

Uniform duality in semi-infinite convex optimization

We study infinite sets of convex functional constraints, with possibly a set constraint, under general background hypotheses which require closed functions and a closed set, but otherwise do not require a Slater point. For example, when the set constraint is not present, only the consistency of the conditions is needed. We provide hypotheses, which are necessary as well as sufficient, for the overall set of constraints to have the property that there is no gap in Lagrangean duality for every convex objective function defined on ℝⁿ. The sums considered for our Lagrangean dual are those involving only finitely many nonzero multipliers. In particular, we recover the usual sufficient condition when only finitely many functional constraints are present. We show that a certain compactness condition in function space plays the role of finiteness, when there are an infinite number of functional constraints. The author's research has been partially supported by Grant ECS8001763 of the National Science Foundation.     

Generating functions and duality for integer programs

We consider the integer program P→max c^′x|Ax=y;xNⁿ . Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables λR^m have discrete analogues zC^m, related in a simple manner. Moreover, both optimal values of P and the LP obey the same formula, using z for P and |z| for the LP. One retrieves (and refines) the so-called group-relaxations of Gomory which, in this dual approach, arise naturally from a detailed analysis of a generalized residue formula of Brion and Vergne. Finally, we also provide an explicit formulation of a dual problem P^*, the analogue of the dual LP in linear programming.     

Conjugate duality for vector-maximization problems

In this article we present a conjugate duality for a problem of maximizing a polyhedral concave nondecreasing homogeneous function over a convex feasible set in the nonnegative n-dimensional orthant. Using this duality we obtain a zero-gap duality for a vector-maximization problem.     

Stability of linearly constrained convex quadratic programs

This paper establishes a simple necessary and sufficient condition for the stability of a linearly constrained convex quadratic program under perturbations of the linear part of the data, including the constraint matrix. It also establishes results on the continuity and differentiability of the optimal objective value of the program as a function of a parameter specifying the magnitude of the perturbation. The results established herein directly generalize well-known results on the stability of linear programs.     

Dynamic convex duality in constrained utility maximization

ABSTRACT

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of forward and backward stochastic differential equations (FBSDEs) plus some additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. We also find that the optimal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problem directly.     

On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type

Sufficient conditions of optimality are derived for convex and non-convex problems with state constraints on the basis of the apparatus of locally conjugate mappings. The duality theorem is formulated and the conditions under which the direct and dual problems are connected by the duality relation are searched for.