Issue 1, Volume 34, April 2009

2009 · Open access

1. Foreword PDF
2. EXISTENCE CONDITIONS FOR QUASIVARIATIONAL INCLUSION PROBLEMS IN $G$-CONVEX SPACES PHAN QUOC KHANH, NGUYEN HONG QUAN PDF

   ▸ Abstract

   We consider general quasivariational inclusion problems in $G$-convex spaces and prove sufficient conditions for the solution existence. The verifiability of the imposed assumptions is discussed in details. Corollaries and examples are supplied to show that the obtained results contain and improve recent existence conditions in the literature, even when applied to particular cases.

   1-10
3. A SOLUTION FOR THE AGGREGATE PRODUCTION PLANNING PROBLEM IN A MULTI-PLANT, MULTI-PERIOD AND MULTI-PRODUCT ENVIRONMENT LORENA PRADENAS, CÉSAR ALVAREZ, JACQUES A. FERLAND PDF

   ▸ Abstract

   In this paper, we introduce a linear mathematical model for the Aggregate Production Planning problem for a forestry company operating several sawmills. Considering different types of raw materials, different production parameters, and a group of sawmills, the industry tries to identify the best levels of production, of subcontracting production, and of inventory to satisfy the demand for different product families in order to maximize its revenue. The linear model is solved efficiently with two linear programming software Cplex 8.1 and Xpress-Optimizer 16.10.02, and numerical results indicate the superiority of the latter.

   11-17
4. ON OUTLIER DETECTION IN MULTIVARIATE TIME SERIES JEAN-MARIE HELBLING, ROBERT CLÉROUX PDF

   ▸ Abstract

   This paper deals with the detection of outliers in multivariate time series models. The different detection methods found in the literature are reviewed and a new one is suggested. It is based on the coefficient of vector autocorrelation. We obtain its influence function which is used in a heuristic for detecting outliers. Then the distribution of the influence function is obtained and used for testing the hypothesis of presence of outliers.

   19-26
5. AN EXISTENCE RESULT FOR MINTY VARIATIONAL INEQUALITIES SUSANA SCHEIMBERG, NELSON MACULAN, MARIA HELENA C. JARDIM PDF

   ▸ Abstract

   In this paper, we establish an existence result of solutions to Minty Variational Inequality under a generalized properly quasimonotone assumption that we introduce. This condition is weaker than those given in the literature for this problem. We illustrate it by an example. We also use this condition to extend the result given by Kassay, Kolumbán and Páles for Minty variational inequality systems.

   27-35
6. MODELS, DATA AND IMAGES FOR PREDICTING THE EVOLUTION OF GEOPHYSICAL FLUIDS FRANCOIS-XAVIER LE DIMET, INNOCENT SOUOPGUI, OLIVIER TITAUD, ARTHUR VIDARD PDF

   ▸ Abstract

   Predicting the evolution of the environment is an important and difficult task. To achieve this goal we need to consider all the available information: mathematical information under the form of models, observations, statistics, images from satellites. We will see how to use optimal control methods in order to link together these sources of information for retrieving at best the state of the environment. A particular attention will be paid to the assimilation of dynamic images in numerical models.

   37-47
7. ACCELERATING CONVERGENCE OF A CLASS OF SPLITTING ALGORITHMS WITH ITERATIVE FOLDINGS ARNAUD LENOIR, PHILIPPE MAHEY PDF

   ▸ Abstract

   We analyze here the asymptotic convergence properties of a class of splitting algorithms for finding a zero of the sum of two maximal monotone operators, not necessarily differentiable. We use the concept of proto- differentiability to obtain some new bounds on the global error of the sequence of iterates and explain the bad spiraling effect already observed in practice. After linking our model with the Lawrence-Spingarn’s folding operators, we show how to accelerate convergence by skipping the averaging steps in a particular way. Numerical results on medium-term stochastic planning problems confirm the nice behavior of the modified algorithm.

   49-65
8. GENERALIZED PROJECTION METHOD FOR NON-LIPSCHITZ MULTIVALUED MONOTONE VARIATIONAL INEQUALITIES PHAM NGOC ANH, LE DUNG MUU, JEAN-JACQUES STRODIOT PDF

   ▸ Abstract

   We generalize the projection method for solving strongly monotone multivalued variational inequalities when the cost operator is not necessarily Lipschitz. At each iteration at most one projection onto the constrained set is needed. When the convex constrained set is not polyhedral, we embed the proposed method in a polyhedral outer approximation procedure. This allows us to obtain the projections by solving strongly convex quadratic programs with linear constraints. We also discuss how to use the proposed method to implement inexact proximal point methods.

   67-79
9. SECOND-ORDER OPTIMALITY CONDITIONS IN SET-VALUED OPTIMIZATION BY A NEW TANGENTIAL DERIVATIVE G. ISAC, A. A. KHAN PDF

   ▸ Abstract

   This paper gives new second-order necessary and sufficient optimality conditions in set-valued optimization. We define second-order tangential derivative/epiderivative of set-valued maps by taking contingent derivative of the first-order contingent derivative. The resulting derivatives/epiderivatives have strikingly simple structure and nice properties. The proposed derivatives are then employed to give new second-order optimality conditions for weakminimality in set-valued optimization.

   81-90
10. IMPLEMENTATION ISSUES FOR HIGH-ORDER ALGORITHMS JEAN-PIERRE DUSSAULT, BENOIT HAMELIN, BILEL KCHOUK PDF

    ▸ Abstract

    Newton-Raphson method, which dates back to 1669–1670, is widely used to solve systems of equations and unconstrained optimization problems. Newton-Raphson consists in linearizing the system of equations and provides quadratic local convergence order. Quite soon after Newton and Raphson introduced their iterative process, Halley in 1694 proposed a higherorder method providing cubic asymptotic convergence order. Chebyshev in 1838 proposed another high-order variant. In High-order Newton-penalty Algorithms [2], by interpreting Newton’s iteration as a linear extrapolation, formulæ were proposed to compute higher-order extrapolations generalizing Newton-Raphson’s and Chebyshev’s methods.

    In this paper, we provide details using an automatic differentiation (AD) tool to implement those high-order extrapolations. We present a complexity analysis allowing to predict the efficiency of those high-order strategies.

    91-103
11. CONVEX METRICS REVISITED F. PLASTRIA, G. SONCK, W. JACQUET PDF

    ▸ Abstract

    This short note gives a new proof and some extensions of the classical result of Witzgall that any convex weak metric is derived from a gauge.

    105-110
12. QUASI-EQUILIBRIUM INCLUSION PROBLEMS OF THE BLUM-OETTLI TYPE AND RELATED PROBLEMS LAI-JIU LIN, NGUYEN XUAN TAN PDF

    ▸ Abstract

    Several quasi-equilibrium inclusion problems of the Blum-Oettli type are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of general vector ideal (resp. proper, Pareto, weak) quasi-optimization problems, of quasivariational inequalities, and of quasivariational inclusion problems.

    111-123
13. QUALIFICATION AND OPTIMALITY CONDITIONS FOR DC PROGRAMS WITH INFINITE CONSTRAINTS N. DINH, B. S. MORDUKHOVICH, T. T. A. NGHIA PDF

    ▸ Abstract

    The paper is devoted to the study of a new class of optimization problems with objectives given as differences of convex (DC) functions and constraints described by infinitely many convex inequalities. We consider such problems in the general framework of locally convex topological vector spaces, although the major results obtained in the paper are new even in finite dimensions when the problems under consideration reduce to DC semi-infinite programs. The main attention is paid to deriving qualified necessary optimality conditions as well as necessary and sufficient optimality conditions for DC infinite and semi-infinite programs and to establishing relations between various qualification conditions. The results obtained are applied to and specified for particular classes of DC programs involving polyhedral convex functions in DC objectives, programs with cone constraints as well as those with positive semi-definite constraints described via the Löwner partial order.

    125-155
14. CODERIVATIVE CALCULATION RELATED TO A PARAMETRIC AFFINE VARIATIONAL INEQUALITY PART 1: BASIC CALCULATIONS J.-C. YAO, N. D. YEN PDF

    ▸ Abstract

    Consider a parametric affine variational inequality $0\in Mx + q + N(x; \Delta(A; b))$; denoted by AVI$(M, q, A, b)$; for which the pair $(q, b) \in \mathbb R^n \times \mathbb R^m$ describes the linear perturbations. Here the matrices $M \in \mathbb R^{n\times n}$ and $A \in \mathbb R^{m\times n}$ are the given data, $\Delta(A, b) = \{x \in \mathbb R^n : Ax \leqslant b\}$ is a polyhedral convex constraint set, and $N(x; \Delta(A, b))$ denotes the normal cone to $\Delta(A, b)$ at $x$. We study the normal coderivative of the normal-cone operator $(x, b) \mapsto N(x; \Delta(A; b))$. In the second part of this paper [20], combining the obtained results with some theorems from Mordukhovich [11], Levy and Mordukhovich [10], Yen and Yao [21], we get sufficient conditions for the Aubin property (the Lipschitz-like property) and the local metric regularity in Robinson’s sense of the solution map $(q, b) \mapsto S(q, b)$ of the problem AVI$(M, q, A, b)$ and of the solution map $(w, b) \mapsto S(w, b)$ of the problem $0 \in f(x, w) + N(x; \Delta(A, b))$ where $f : \mathbb R^n \times \mathbb R^s \to \mathbb R^n$ is a given $C^1$ vector function. Our investigation complements the well-known work of Dontchev and Rockafellar [3] where the Aubin property of the solution maps $q \mapsto S(q, b)$ and $w \mapsto S(w, b)$ ($b$ is fixed) was established via a critical face condition.

    157-172