Robust conjugate duality for convex optimization under uncertainty with application to data classification

In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition. This regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem. More importantly, we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem. As an application, we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets. The support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences.     

Robust duality for generalized convex programming problems under data uncertainty

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. A robust strong duality theorem is given whenever the Lagrangian function is convex. We provide classes of uncertain non-convex programming problems for which robust strong duality holds under a constraint qualification. In particular, we show that robust strong duality is guaranteed for non-convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized Slater condition. Numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems. We further show that robust duality continues to hold under a weakened convexity condition.     

Necessary and sufficient conditions for Kuhn-Tucker type optimality and for weak duality of nonsmooth programming

In this paper, we are concerned with a nonsmooth programming problem with inequality constraints. We obtain an optimality condition for Kuhn-Tucker points to be minimizers. Later on, we present necessary and sufficient conditions for weak duality between the primal problem and its mixed type dual, which help us to extend some earlier work from the literature.     

Duality for nondifferentiable minimax programming in complex spaces

Employing the optimality (necessary and sufficient) conditions of a nondifferentiable minimax programming problem in complex spaces, we formulate a one-parametric dual and a parameter free dual problems. On both dual problems, we establish three duality theorems: weak, strong, and strict converse duality theorem, and prove that there is no duality gap between the two dual problems with respect to the primal problem under some generalized convexities of complex functions in the complex programming problem.     

An analysis of some dual problems in multiobjective optimization (I)

In this work we study the duality for a general multiobjective optimization problem. Considering, first, a scalar problem, different duals using the conjugacy approach are presented. Starting from these scalar duals, we introduce six different multiobjective dual problems to the primal one, one depending on certain vector parameters. The existence of weak and, under certain conditions, strong duality between the primal and the dual problems is shown. Afterwards, some inclusion results for the image sets of the multiobjective dual problems (D ₁), (D _α) and (D_FL ) are derived. Moreover, we verify that the efficiency sets within the image sets of these problems coincide, but the image sets themselves do not.     

An algorithm for semi-infinite transportation problems

In this paper we consider a class of semi-infinite transportation problems. We develop an algorithm for this class of semi-infinite transportation problems. The algorithm is a primal dual method which is a generalization of the classical algorithm for finite transportation problems. The most important aspect of our paper is that we can prove the convergence result for the algorithm. Finally, we implement some examples to illustrate our algorithm.     

Unconstrained duals to partially separable constrained programs

In this note partially separable convex programs are dualized in such a way that, under certain assumptions, unconstrained concave duals arise. A return formula is given by which the solution of the primal is directly computed if a solution of the dual is known. Further, the solvability of both the primal and the dual is shown to depend essentially on the behaviour of the lower dimensional programs for determining the Fenchel conjugates.     

Minimizing measures of risk by saddle point conditions

The minimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions. This new problem involves Banach spaces but a mean value theorem for risk measures is stated, and this simplifies the dual problem. Then, optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable. This characterization is significantly different if one compares it with previous literature. Furthermore, the saddle point condition very easily applies in practice. Four applications in finance and insurance are presented.     

Dual versus primal-dual interior-point methods for linear and conic programming

We observe a curious property of dual versus primal-dual path-following interior-point methods when applied to unbounded linear or conic programming problems in dual form. While primal-dual methods can be viewed as implicitly following a central path to detect primal infeasibility and dual unboundedness, dual methods can sometimes implicitly move away from the analytic center of the set of infeasibility/unboundedness detectors. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

Saddle points and gap functions for vector equilibrium problems via conjugate duality in vector optimization

In this paper, two conjugate dual problems based on weak efficiency to a constrained vector optimization problem are introduced. Some inclusion relations between the dual objective mappings and the properties of the Lagrangian maps and their saddle points for primal problem are discussed. Gap functions for a vector equilibrium problem are established by using the weak and strong duality.     

Partially finite convex programming,Part II: Explicit lattice models

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L ¹ approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.     

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 the finite extension of the marginal function arising in decomposition algorithms 1

We consider the problem how a convex optimal-value function arising in primal decomposition can be finitely continued beyond its domain. By a suitable presentation of the exact penalty method an implementable continuation can be obtained which does not change the set of optimal solutions. If the problem has separability and partially linearity properties we manage to obtain a complete continuation of the optimal-value function.     

A characterization of pseudoinvexity for the efficiency in non-differentiable multiobjective problems. Duality

We prove that in order for the Kuhn-Tucker or Fritz John points to be efficient solutions, it is necessary and sufficient that the non-differentiable multiobjective problem functions belong to new classes of functions that we introduce here: KT-pseudoinvex-II or FJ-pseudoinvex-II, respectively. We illustrate it by examples. These characterizations generalize recent results given for the differentiable case. We study the dual problem and establish weak, strong and converse duality results.     

Strong Duality with Proper Efficiency in Multiobjective Optimization Involving Nonconvex Set-Valued Maps

In this paper, we consider some dual problems of a primal multiobjective problem involving nonconvex set-valued maps. For each dual problem, we give conditions under which strong duality between the primal and dual problems holds in the sense that, starting from a Benson properly efficient solution of the primal problem, we can construct a Benson properly efficient solution of the dual problem such that the corresponding objective values of both problems are equal. The notion of generalized convexity of set-valued maps we use in this paper is that of near-subconvexlikeness.     

A modified reduced gradient method for a class of nondifferentiable problems

The class of nondifferentiable problems treated in this paper constitutes the dual of a class of convex differentiable problems. The primal problem involves faithfully convex functions of linear mappings of the independent variables in the objective function and in the constraints. The points of the dual problem where the objective function is nondifferentiable are known: the method presented here takes advantage of this fact to propose modifications necessary in the reduced gradient method to guarantee convergence.     

Generalized convex relations with applications to optimization and models of economic dynamics

We examine a notion of generalized convex set-valued mapping, extending the notions of a convex relation and a convex process. Under general conditions, we establish duality results for composite set-valued mappings and for convex programming problems involving convex set-valued mappings. We also present applications to the study of economic dynamical systems, by obtaining the characteristics of optimal paths generated by convex processes, and to optimization problems of a certain class of positively homogeneous increasing functions.     

Convex optimization problems with arbitrary right-hand side perturbations

The problem of finding a solution to a system of mixed variational inequalities, which can be interpreted as a generalization of a primal–dual formulation of an optimization problem under arbitrary right-hand side perturbations, is considered. A number of various equilibrium type problems are particular cases of this problem. We suggest the problem to be reduced to a class of variational inequalities and propose a general descent type method to find its solution. If the primal cost function does not possess strengthened convexity properties, this descent method can be combined with a partial regularization method.     

A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.Corresponding author. Partially supported by Air Force Office of Scientific Research Grant 91-0008 and National Science Foundation Grant DMS-9201297.     

A primal dual modified subgradient algorithm with sharp Lagrangian

We apply a modified subgradient algorithm (MSG) for solving the dual of a nonlinear and nonconvex optimization problem. The dual scheme we consider uses the sharp augmented Lagrangian. A desirable feature of this method is primal convergence, which means that every accumulation point of a primal sequence (which is automatically generated during the process), is a primal solution. This feature is not true in general for available variants of MSG. We propose here two new variants of MSG which enjoy both primal and dual convergence, as long as the dual optimal set is nonempty. These variants have a very simple choice for the stepsizes. Moreover, we also establish primal convergence when the dual optimal set is empty. Finally, our second variant of MSG converges in a finite number of steps.