Lipschitz-like property of an implicit multifunction and its applications

The aim of this work is twofold. First, we use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of an implicit multifunction. More explicitly, new sufficient conditions in terms of the Fréchet coderivative and the normal/Mordukhovich coderivative of parametric multifunctions for this implicit multifunction to have the Lipschitz-like property at a given point are established. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map in parametric vector optimization problems by employing the above implicit multifunction results.     

Clarke coderivatives of efficient point multifunctions in parametric vector optimization

The paper is devoted to the study of the Clarke/circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in Banach spaces. We provide inner/outer estimates for evaluating the Clarke/circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical, operator and (finite and infinite) functional constraints. Examples are given for analyzing and illustrating the obtained results.     

Normal subdifferentials of efficient point multifunctions in parametric vector optimization

The normal subdifferential of a set-valued mapping with values in a partially ordered Banach space has been recently introduced in Bao and Mordukhovich (Control Cyber 36:531–562, 2007), by using the Mordukhovich coderivative of the associated epigraphical multifunction, which has proven to be useful in deriving necessary conditions for super efficient points of vector optimization problems. In this paper, we establish new formulae for computing and/or estimating the normal subdifferential of the efficient point multifunctions of parametric vector optimization problems. These formulae will be presented in a broad class of conventional vgector optimization problems with the presence of geometric, operator, equilibrium, and (finite and infinite) functional constraints.     

Implicit Multifunction Theorems

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson–Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson–Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.     

Derivatives of the Efficient Point Multifunction in Parametric Vector Optimization Problems

In this paper, we deal with the sensitivity analysis in vector optimization. More specifically, formulae for inner and outer evaluating the S-derivative of the efficient point multifunction in parametric vector optimization problems are established. These estimating formulae are presented via the set of efficient/weakly efficient points of the S-derivative of the original multifunction, a composite multifunction of the objective function and the constraint mapping. The elaboration of the formulae in vector optimization problems, having multifunction constraints and semiinfinite constraints, is also undertaken. Furthermore, examples are provided for analyzing and illustrating the obtained results.     

Point-based sufficient conditions for metric regularity of implicit multifunctions

We obtain some point-based sufficient conditions for the metric regularity in Robinson’s sense of implicit multifunctions in a finite-dimensional setting. The new implicit function theorem (which is very different from the preceding results of Ledyaev and Zhu [Yu.S. Ledyaev, Q.J. Zhu, Implicit multifunctions theorems, Set-Valued Anal. 7 (1999) 209–238], Ngai and Théra [H.V. Ngai, M. Théra, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal. 12 (2004) 195–223], Lee, Tam and Yen [G.M. Lee, N.N. Tam, N.D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008) 11–22]) can be used for analyzing parametric constraint systems as well as parametric variational systems. Our main tools are the concept of normal coderivative due to Mordukhovich and the corresponding theory of generalized differentiation.     

Characterizing the Single-Valuedness of Multifunctions

We characterize the local single-valuedness and continuity of multifunctions (set-valued mappings) in terms of their premonotonicity and lower semicontinuity. This result completes the well-known fact that lower semicontinuous, monotone multifunctions are single-valued and continuous. We also show that a multifunction is actually a Lipschitz single-valued mapping if and only if it is premonotone and has a generalized Lipschitz property called Aubin continuity. The possible single-valuedness and continuity of multifunctions is at the heart of some of the most fundamental issues in variational analysis and its application to optimization. We investigate the impact of our characterizations on several of these issues; discovering exactly when certain generalized subderivatives can be identified with classical derivatives, and determining precisely when solutions to generalized variational inequalities are locally unique and Lipschitz continuous. As an application of our results involving generalized variational inequalities, we characterize when the Karush–Kuhn–Tucker pairs associated with a parameterized optimization problem are locally unique and Lipschitz continuous.     

On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Generalized Equation

The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic generalized equation. It is demonstrated that, under suitable conditions, both the cosmic deviation and the ρ-deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic generalized equation converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric generalized equations is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints.     

Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints

We consider a new class of optimization problems involving stochastic dominance constraints of second order. We develop a new splitting approach to these models, optimality conditions and duality theory. These results are used to construct special decomposition methods.This research was supported by the NSF awards DMS-0303545 and DMS-0303728.Key words.Stochastic programming – stochastic ordering – semi-infinite optimized – decomposition     

Globally Optimized Spherical Point Arrangements: Model Variants and Illustrative Results

The well-balanced distribution of points over the surface of a sphere is of significant interest in various fields of science. The quality of point configurations is typically expressed by criterion functions that have many local optima. A general global optimization framework is suggested to solve such problems. To illustrate the viability of this approach, the model development and solver system LGO is applied to four different model versions. Numerical results – including the visual representation of criterion functions in these models – are presented. The global optimization approach can be tailored to specific problem settings, and it is also applicable to a large variety of other model forms.     

Openness properties for parametric set-valued mappings and implicit multifunctions

The aim of this paper is to obtain some openness results in terms of normal coderivative for parametric set-valued mappings acting between infinite dimensional spaces. Then, implicit multifunction results are obtained by simply specializing the openness results. Moreover, we study a kind of metric regularity of the implicit multifunction. The results of the paper generalize several recent results in literature.     

Sensitivity Analysis of Multiobjective Optimization Problems with Parameterized Quasi-Variational Inequalities

This paper mainly establishes the sensitivity analysis of a multiobjective optimization problem with parameterized quasi-variational inequalities (QVIs). Using the (regular) coderivative of the associated epigraphical multifunction, the (regular) subdifferentials of the efficient frontier maps are estimated, which involve the (regular) coderivatives of the solution mapping to the parameterized QVIs. Under the linear independent constraint qualification, the defined auxiliary set-valued mappings in the parameterized QVIs are clam. The detailed formulae of subdifferentials of the efficient frontier maps are obtained and examples are simultaneously provided for analyzing and illustrating the obtained results.     

A note on implicit multifunction theorems

The paper is mainly devoted to the study of implicit multifunction theorems in terms of Fréchet coderivative in Asplund spaces. It sharpens the well-known implicit multifunction theorem of Ledyaev and Zhu (Set Valued Anal., 7, 209–238, 1999) as well as many recent publications about this significant area.     

Singular stationary measures are not always fractal

We know of few explicit results to insure that stationary measures are simultaneously (i) singular, (ii) nonatomic, (iii) with interval support, and (iv) unique. Such results would appear useful, to further separate the analytic notion of singular from the geometric notion of fractal. We prove two general theorems, one for maps of [0,1] into [0, 1], the other for 2×2 random matrices. In each setting, we study measures supported on two points of the transformation space, and we provide sufficient conditions to insure that the stationary measures satisfy (i)–(iv).     

On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Variational Inequality

The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic variational inequality with equality and inequality constraints. The notion of integrated deviation is introduced to characterize the outer limit of a sequence of sets. It is demonstrated that, under suitable conditions, both the cosmic deviation and the integrated deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic variational inequality converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric stochastic variational inequality is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic bilevel program.     

On Rapidly Convergent Series Expressions for Zeta- and L-Values,and Log Sine Integrals

Regarding the rapidly convergent series expansion for special values of - and L-functions for integer points, there are two approaches.One approach starts from Euler's 1772 formula for (3) and culminates in Srivastava's very recent results via many intermediate results, and the other is due to Wilton's investigation, which was shown by us (Aeq. Math. 59, 2000, 1–19) to be a consequence of Ramanujan's work (Collected Papers of Srinivasa Ramanujan, CUP 1927, reprint Chelsea, 1962, pp. 163–168).More recently, Katsurada (Acta Arith. 90, 1999, 79–89.) has generalized all existing formulas into a rather wide framework of Dirichlet L-functions.Our purpose is to show that even the most general Katsurada's formulas are easy consequences of our fundamental summation formulas for the series with Hurwitz zeta-function coefficients.We give a three-line proof of Katsurada's main theorem, and also we make some remarks on the recent paper of Bradley (The Ramanujan J. 3, 1999, 159–173).     

Local search algorithms for the rectangle packing problem with general spatial costs

We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms. Key words.rectangle packing – sequence pair – general spatial cost – dynamic programming – metaheuristicsMathematics Subject Classification (1991):20E28, 20G40, 20C20     

A Stone–Weierstrass Type Theorem for an Unstructured Set – with Applications

A general version of the Stone–Weierstrass theorem is presented – one which involves no structure on the domain set of the real valued functions. This theorem is similar to the Stone–Weierstrass theorem which appears in the book by Gillman and Jerison, but instead of involving the concept of stationary sets the one presented here involves stationary filters. As a corollary to our results we obtain Nel's theorem of Stone–Weierstrass type for an arbitrary topological space. Finally, an application is made to the setting of Cauchy spaces.     

Minimal concave cost rebalance of a portfolio to the efficient frontier

One usually constructs a portfolio on the efficient frontier, but it may not be efficient after, say three months since the efficient frontier will shift as the elapse of time. We then have to rebalance the portfolio if the deviation is no longer acceptable. The method to be proposed in this paper is to find a portfolio on the new efficient frontier such that the total transaction cost required for this rebalancing is minimal. This problem results in a nonconvex minimization problem, if we use mean-variance model. In this paper we will formulate this problem by using absolute deviation as the measure of risk and solve the resulting linearly constrained concave minimization problem by a branch and bound algorithm successfully applied to portfolio optimization problem under concave transaction costs. It will be demonstrated that this method is efficient and that it leads to a significant reduction of transaction costs. Key words.portfolio optimization – rebalance – mean-absolute deviation model – concave cost minimization – optimization over the efficient set – global optimizationMathematics Subject Classification (1991):20E28, 20G40, 20C20