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In solving certain optimization problems, the corresponding Lagrangian dual problem is often solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that under certain conditions the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. Another interesting aspect of dual problems is that both lower and upper-level optimization problems involve only box constraints and no other equality of inequality constraints. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for co-evolving two populations - one involving Lagrange multipliers and other involving decision variables - to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested smooth and nonsmooth algorithms and a couple of previously suggested coevolutionary algorithms. The performance of CEDO algorithm is also compared with two classical methods involving nonsmooth (bundle) optimization methods. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed coevolutionary algorithm compared to usual nested smooth and nonsmooth algorithms and other existing coevolutionary approaches remain as the hallmark of the current study. © 2012 Springer Science+Business Media New York.

World University root node

School of Artificial Intelligence and Data Science (AIDE)

Journal of Global Optimization

Solving dual problems using a coevolutionary optimization algorithm

Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is "No" even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lower-level problem satisfies the Slater's constraint qualification. We also show by examples that this correspondence can fail if the Slater's constraint qualification fails to hold at lower-level. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples. © 2010 Springer and Mathematical Programming Society.

Mathematical Programming

Is bilevel programming a special case of a mathematical program with complementarity constraints?

In this article we introduce the notion of approximate Karush-Kuhn- Tucker (KKT) points for smooth, convex and nonsmooth, nonconvex vector optimization problems. We study a kind of stability of these points and KKT points of vector optimization problems. In the convex case we also introduce and study the notion of modified approximate KKT points motivated by Ekeland's variational principle. We prove stability properties of these points for several optimization problems. © 2011 Taylor & Francis.

Optimization

Stability properties of KKT points in vector optimization

A numerical technique is presented for constructing an approximation of the weak Pareto front of nonconvex multi-objective optimization problems, based on a new Tchebychev-type scalarization and its equivalent representations. First, existing results on the standard Tchebychev scalarization, the weak Pareto and Pareto minima, as well as the uniqueness of the optimal value in the Pareto front, are recalled and discussed for the case when the set of weak Pareto minima is the same as the set of Pareto minima, namely, when weak Pareto minima are also Pareto minima. Of the two algorithms we present, Algorithm 1 is based on this discussion. Algorithm 2, on the other hand, is based on the new scalarization incorporating rays associated with the weights of the scalarization in the value (or objective) space, as constraints. We prove two relevant results for the new scalarization. The new scalarization and the resulting Algorithm 2 are particularly effective in constructing an approximation of the weak Pareto sections of the front. We illustrate the working and capability of both algorithms by means of smooth and nonsmooth test problems with connected and disconnected Pareto fronts. © 2011 Copyright Taylor and Francis Group, LLC.

Fulltext

A new scalarization and numerical method for constructing the weak Pareto front of multi-objective optimization problems

Gap functions have proved to be efficient tools to study single-valued variational inequalities. This approach allows us to reformulate the problem into an optimization problem. New notions of gap functions are defined for set-valued variational inequalities. We prove finiteness and error bounds properties, i.e. upper estimates for the distance to the solution set of the variational inequality. © 2011 Springer Science+Business Media, LLC.

Journal of Optimization Theory and Applications

On Gap Functions for Multivalued Stampacchia Variational Inequalities

In this paper we introduce a scalar gap function for a Stampacchia type vector variational inequality problem. We show how properties of the gap function can be improved by regularizing it in the sense of Fukushima. We study in detail the properties of this regularized gap function. An error bound for the vector variational inequality is obtained in terms of the regularized gap function. We also introduce the notion of D-gap function for vector variational inequalities. We study its properties and also present an error bound in terms of the D-gap functions. © 2010 Yokohama Publishers.

Pacific Journal of Optimization

Regularized gap functions and error bounds for vector variational inequalities

We present a new family of proximal point methods for solving monotone variational inequalities. Our algorithm has a relative error tolerance criterion in solving the proximal subproblems. Our convergence analysis covers a wide family of regularization functions, including double regularizations recently introduced by Silva, Eckstein, and Humes, Jr.[SIAM J. Optim., 12 (2001), pp. 238-261] and the Bregman distance induced by h(x) = ∑i=1 n xi log xi We do not use in our analysis the assumption of paramonotonicity, which is standard in proving convergence of Bregman-based proximal methods. © by SIAM.

SIAM Journal on Optimization

Inexact proximal point methods for variational inequality problems

The problem to find a best solution within the set of optimal solutions of a convex optimization problem is modeled as a bilevel programming problem. It is shown that regularity conditions like Slater’s constraint qualification are never satisfied for this problem. If the lower-level problem is replaced with its (necessary and sufficient) optimality conditions, it is possible to derive a necessary optimality condition for the resulting problem. An example is used to show that this condition in not sufficient even if the initial problem is a convex one. If the lower-level problem is replaced using its optimal value, it is possible to obtain an optimality condition that is both necessary and sufficient in the convex case. © Springer Science+Business Media, LLC 2010.

Springer Optimization and Its Applications

Optimality conditions for a simple convex bilevel programming problem

Evolutionary algorithms (EAs) are increasingly being applied to solve real-parameter optimization problems due to their flexibility in handling complexities such as non-convexity, non-differentiability, multi-modality and noise in problems. However, an EA's solution is never guaranteed to be optimal in generic problems, even for smooth problems, and importantly EAs still lack a theoretically motivated termination criterion for stopping an EA run only when a near-optimal point is found. We address both these issues in this paper by integrating the Karush-Kuhn-Tucker (KKT) optimality conditions that involve first-order derivatives of objective and constraint functions with an EA. For this purpose, we define a KKT-proximity measure by relaxing the complimentary slackness condition associated with the KKT conditions. Results on a number of standard constrained test problems indicate that in spite of not using any gradient information and any theoretical optimality conditions, an EA's selection, recombination and mutation operation lead the search process to a point close to the KKT point. This suggests that the proposed KKT-proximity measure can be used a termination criterion in an EA simulation. Copyright 2010 ACM.

Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference, GECCO '10

Investigating EA solutions for approximate KKT conditions in smooth problems

We consider unconstrained minimax problem where the objective function is the maximum of a finite number of smooth convex functions. We present an iterative method to compute the optimal solution for the unconstrained convex finite minimax problem. The algorithm developed estimates the direction of steepest-descent rapidly and using Armijo's condition proceeds towards the solution. Owing to the highly parallel nature of the algorithm, it is highly suitable for large minimax problems. Algorithm is implemented on Nvidia Tesla C1060 graphics card using CUDA and numerical comparisons with RGA & CFSQP are presented. © 2010 Springer-Verlag.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

A parallel algorithm for solving large convex minimax problems

In this note we show that for a large class of optimization problems, the Lagrange multiplier rule can be derived from the approximate multiplier rule. © 2009 Springer and Mathematical Programming Society.

A note on an approximate lagrange multiplier rule

This paper presents some results concerning the existence of the Lagrange multipliers for vector optimization problems in the case where the ordering cone in the codomain has an empty interior. The main tool for deriving our assertions is a scalarization by means of a functional introduced by Hiriart-Urruty (Math. Oper. Res. 4:79-97, 1979) (the so-called oriented distance function). Moreover, we explain some applications of our results to a vector equilibrium problem, to a vector control-approximation problem and to an unconstrainted vector fractional programming problem. © Springer Science+Business Media, LLC 2009.

Lagrange multipliers for ε-pareto solutions in vector optimization with nonsolid cones in banach spaces

This paper studies the class of increasing and co-radiant (ICR) functions over a cone equipped with an order relation which agrees with the conic structure. In particular, a representation of ICR functions as abstract convex functions is provided. This representation suggests the introduction of some polarity notions between sets. The relationship between ICR functions and increasing positively homogeneous functions is also shown.

Journal of Convex Analysis

Monotonic analysis over cones: III

The aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well. © 2008 Elsevier Inc. All rights reserved.

Journal of Mathematical Analysis and Applications

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

In this paper our focus is to study necessary optimality conditions for nonsmooth vector optimization problems in Banach spaces. Since in most infinite dimensional spaces the natural ordering cone has an empty interior it becomes quiet difficult to study vector optimization problems. This is due to the fact that we do not have powerful scalarization techniques in such a situation. In this paper we show how we can overcome such difficulties by using the notion of dilating cones. © 2008 Yokohama Publishers.

Lagrange multipliers for pareto minima in general banach spaces

It is well known that the generalized Nash equilibrium problem, a model for multi-leader-follower games, can be reformulated as a quasivariational inequality. We show that, in fact, a reformulation in terms of a variational inequality can be obtained in the general setting of quasiconvex nondifferentiable decision functions. An existence result is deduced. © 2008 Elsevier B.V. All rights reserved.

Operations Research Letters

Generalized Nash equilibrium problem, variational inequality and quasiconvexity

Despite having a wide-spread applicability of evolutionary optimization procedures over the past few decades, EA researchers still face criticism about the theoretical optimality of obtained solutions. In this paper, we address this issue for problems for which gradients of objectives and constraints can be computed either exactly, or numerically or through subdifferentials. We suggest a systematic procedure of analyzing a representative set of Pareto-optimal solutions for their closeness to satisfying Karush-Kuhn-Tucker (KKT) points, which every Pareto-optimal solution must also satisfy. The procedure involves either a least-square solution or an optimum solution to a set of linear system of equations involving Lagrange multipliers. The procedure is applied to a number of differentiable and non-differentiable test problems and to a highly nonlinear engineering design problem. The results clearly show that EAs are capable of finding solutions close to theoretically optimal solutions in various problems. As a by-product, the error metric suggested in this paper can also be used as a termination condition for an EA application. Hopefully, this study will bring EAs and its research closer to classical optimization studies. © 2007 IEEE.

2007 IEEE Congress on Evolutionary Computation, CEC 2007

Finding trade-off solutions close to KKT points using evolutionary multi-objective optimization

The article is devoted to the study of the so-called optimistic version of bilevel programming in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation. Considering a basic optimistic model in bilevel programming, we reduce it to a one-level framework of nondifferentiable programs formulated via (nonsmooth) optimal value function of the parametric lower-level problem in the original model. Using advanced formulas for computing basic subgradients of value/marginal functions in variational analysis, we derive new necessary optimality conditions for bilevel programs reflecting significant phenomena that have never been observed earlier. In particular, our optimality conditions for bilevel programs do not depend on the partial derivatives with respect to parameters of the smooth objective function in the parametric lower-level problem. We present efficient implementations of our approach and results obtained for bilevel programs with differentiable, convex, linear, and Lipschitzian functions describing the initial data of the lower-level and upper-level problems.

New necessary optimality conditions in optimistic bilevel programming

In this article we discuss the conditions required to guarantee the non-emptiness and the boundedness of certain subsets of the set of Lagrange multipliers for an inequality and equality constrained vector minimization problem.

Bounded sets of KKT multipliers in vector optimization

Focus in the article is on necessary optimality conditions for bilevel programming problems. We first identify approaches that seem to be promising. Then, two such approaches are investigated. Using the nonexistence of a descent direction for the objective function within the tangent cone to the feasible set as necessary optimality condition we need efficient ways to describe this tangent cone. We describe possibilities for this in three different special cases. We conclude this article with the second approach applying Mordukhovich's coderivative to an appropriately reformulated bilevel programming problem. © 2006 Taylor & Francis.