Optimality Conditions for Metrically Consistent Approximate Solutions in Vector Optimization

In this paper, approximate solutions of vector optimization problems are analyzed via a metrically consistent ε-efficient concept. Several properties of the ε-efficient set are studied. By scalarization, necessary and sufficient conditions for approximate solutions of convex and nonconvex vector optimization problems are provided; a characterization is obtained via generalized Chebyshev norms, attaining the same precision in the vector problem as in the scalarization. This research was partially supported by the Ministerio de Educación y Ciencia (Spain), Project MTM2006-02629 and by the Consejería de Educación de la Junta de Castilla y León (Spain), Project VA027B06. The authors are grateful to the anonymous referees for helpful comments and suggestions.     

Using scenario trees and progressive hedging for stochastic inventory routing problems

The Stochastic Inventory Routing Problem is a challenging problem, combining inventory management and vehicle routing, as well as including stochastic customer demands. The problem can be described by a discounted, infinite horizon Markov Decision Problem, but it has been showed that this can be effectively approximated by solving a finite scenario tree based problem at each epoch. In this paper the use of the Progressive Hedging Algorithm for solving these scenario tree based problems is examined. The Progressive Hedging Algorithm can be suitable for large-scale problems, by giving an effective decomposition, but is not trivially implemented for non-convex problems. Attempting to improve the solution process, the standard algorithm is extended with locking mechanisms, dynamic multiple penalty parameters, and heuristic intermediate solutions. Extensive computational results are reported, giving further insights into the use of scenario trees as approximations of Markov Decision Problem formulations of the Stochastic Inventory Routing Problem.     

Relaxation of constraints in lexicographic multiobjective programming problems

In this paper, the constraints of the sequential optimization of the lexicographic approach are relaxed in order to obtain a simple alternative approach to solve multiobjective problems. In other words, we allow the decision-maker to deviate from the optimal solution of each iteration if he/she prefers. Some properties of the obtained solutions are studied and the relationship between these solutions and those of the weighted sum scalarization and the ?-constraint scalarization and also the elastic ?-constraint scalarization are investigated. Finally, some examples are provided to show more details.     

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A general approach for studying duality in multiobjective optimization

A general duality framework in convex multiobjective optimization is established using the scalarization with K-strongly increasing functions and the conjugate duality for composed convex cone-constrained optimization problems. Other scalarizations used in the literature arise as particular cases and the general duality is specialized for some of them, namely linear scalarization, maximum (-linear) scalarization, set scalarization, (semi)norm scalarization and quadratic scalarization.      

The algebraic degree of geometric optimization problems

In this paper we apply Galois methods to certain fundamentalgeometric optimization problems whose exact computational complexity has been an open problem for a long time. In particular we show that the classic Weber problem, along with theline-restricted Weber problem and itsthree-dimensional version are in general not solvable by radicals over the field of rationals. One direct consequence of these results is that for these geometric optimization problems there existsno exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction ofkth roots. This leaves only numerical or symbolic approximations to the solutions, where the complexity of the approximations is shown to be primarily a function of the algebraic degree of the optimum solution point.     

Existence and Optimality Conditions for Approximate Solutions to Vector Optimization Problems

In this paper, we introduce a new concept of ϵ-efficiency for vector optimization problems. This extends and unifies various notions of approximate solutions in the literature. Some properties for this new class of approximate solutions are established, and several existence results, as well as nonlinear scalarizations, are obtained by means of the Ekeland’s variational principle. Moreover, under the assumption of generalized subconvex functions, we derive the linear scalarization and the Lagrange multiplier rule for approximate solutions based on the scalarization in Asplund spaces.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

Simulating events of unknown probabilities via reverse time martingales

Let s∈(0,1) be uniquely determined but only its approximations can be obtained with a finite computational effort. Assume one aims to simulate an event of probability s. Such settings are often encountered in statistical simulations. We consider two specific examples. First, the exact simulation of non‐linear diffusions ([ 3 ]). Second, the celebrated Bernoulli factory problem ([ 10 , 13 ]) of generating an f(p)‐coin given a sequence X₁,X₂,… of independent tosses of a p‐coin (with known f and unknown p). We describe a general framework and provide algorithms where this kind of problems can be fitted and solved. The algorithms are straightforward to implement and thus allow for effective simulation of desired events of probability s. Our methodology links the simulation problem to existence and construction of unbiased estimators. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 441–452, 2011     

On Markovian Multi-Class,Multi-Server Queueing

Multi-class multi-server queueing problems are a generalisation of the well-known M/M/k queue to arrival processes with clients of N types that require exponentially distributed service with different average service times. In this paper, we give a procedure to construct exact solutions of the stationary state equations using the special structure of these equations. Essential in this procedure is the reduction of a part of the problem to a backward second order difference equation with constant coefficients. It follows that the exact solution can be found by eigenmode decomposition. In general eigenmodes do not have a simple product structure as one might expect intuitively. Further, using the exact solution, all kinds of interesting performance measures can be computed and compared with heuristic approximations (insofar available in the literature). We provide some new approximations based on special multiplicative eigenmodes, including the dominant mode in the heavy traffic limit. We illustrate our methods with numerical results. It turns out that our approximation method is better for higher moments than some other approximations known in the literature. Moreover, we demonstrate that our theory is useful to applications where correlation between items plays a role, such as spare parts management.     

A logarithmic-quadratic proximal point scalarization method for multiobjective programming

We present a proximal point method to solve multiobjective programming problems based on the scalarization for maps. We build a family of convex scalar strict representations of a convex map F from R ⁿ   to  R ^m with respect to the lexicographic order on R ^m and we add a variant of the logarithmic-quadratic regularization of Auslender, where the unconstrained variables in the domain of F are introduced in the quadratic term. The nonegative variables employed in the scalarization are placed in the logarithmic term. We show that the central trajectory of the scalarized problem is bounded and converges to a weak pareto solution of the multiobjective optimization problem.     

Scalarization of Levitin–Polyak well-posed set optimization problems

The aim of this paper is to study Levitin–Polyak (LP in short) well-posedness for set optimization problems. We define the global notions of metrically well-setness and metrically LP well-setness and the pointwise notions of LP well-posedness, strongly DH-well-posedness and strongly B-well-posedness for set optimization problems. Using a scalarization function defined by means of the point-to-set distance, we characterize the LP well-posedness and the metrically well-setness of a set optimization problem through the LP well-posedness and the metrically well-setness of a scalar optimization problem, respectively.     

New optimality conditions and a scalarization approach for a nonconvex semi-vectorial bilevel optimization problem

In this paper, we are concerned with the optimistic formulation of a semivectorial bilevel optimization problem. Introducing a new scalarization technique for multiobjective programs, we transform our problem into a scalar-objective optimization problem by means of the optimal value reformulation and establish its theoretical properties. Detailed necessary conditions, to characterize local optimal solutions of the problem, were then provided, while using the weak basic CQ together with the generalized differentiation calculus of Mordukhovich. Our approach is applicable to nonconvex problems and is different from the classical scalarization techniques previously used in the literature and the conditions obtained are new.

     

On Approximate Solutions in Vector Optimization Problems Via Scalarization

This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze and widely used in the literature to study this kind of solutions. Necessary and sufficient conditions for Kutateladze’s approximate solutions are given through scalarization, in such a way that these points are approximate solutions for a scalar optimization problem. Necessary conditions are obtained by using gauge functionals while monotone functionals are considered to attain sufficient conditions. Two properties are then introduced to describe the idea of parametric representation of the approximate efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate solutions in convex and nonconvex vector optimization problems are proved and the obtained results are applied to Pareto problems. AMS Classification:90C29, 49M37 This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.     

Efficiency and Solution Approaches to Bicriteria Nonconvex Programs

A new nonlinear scalarization specially designed for bicriteria nonconvexprogramming problems is presented. The scalarization is based on generalizedLagrangian duality theory and uses an augmented Lagrange function. The newconcepts, q _i-approachable points and augmented duality gap, are introducedin order to determine the location of nondominated solutions with respect to aduality gap as well as the connectedness of the nondominated set.     

Scalarization of the fuzzy optimization problems

Scalarization of the fuzzy optimization problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Two solution concepts are proposed by considering two convex cones. The set of all fuzzy numbers can be embedded into a normed space. This motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to solve the fuzzy optimization problems. By applying scalarization to the optimization problem with fuzzy coefficients, we obtain its corresponding scalar optimization problem. Finally, we show that the optimal solution of its corresponding scalar optimization problem is the optimal solution of the original fuzzy optimization problem.     

A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. In this paper we introduce a nonlinear scalarization function for a variable domination structure. Several important properties, such as subadditiveness and continuity, of this nonlinear scalarization function are established. This nonlinear scalarization function is applied to study the existence of solutions for generalized quasi-vector equilibrium problems. This paper is dedicated to Professor Franco Giannessi for his 68th birthday     

Regularization of Ill-Posed Problems by Envelope Guided Conjugate Gradients

Abstract

We propose a new way to iteratively solve large scale ill-posed problems by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain, iteratively, approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired solution can be reconstructed with a low number of iterations. We apply the technique to an idealized image reconstruction problem in positron emission tomography.     

A new alternating‐direction finite element method for hyperbolic equation

A modified backward difference time discretization is presented for Galerkin approximations for nonlinear hyperbolic equation in two space variables. This procedure uses a local approximation of the coefficients based on patches of finite elements with these procedures, a multidimensional problem can be solved as a series of one‐dimensional problems. Optimal order H₀¹ and L² error estimates are derived. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007