Abstract: Discretizations of infinite-dimensional variational inequalities lead to linear and nonlinear complementarity problems with many degrees of freedom. To solve these problems in a parallel computing environment, we propose two active-set methods that solve only one linear system of equations per iteration. The linear solver, preconditioner, and matrix structures can be chosen by the user for a particular application to achieve high parallel performance. The parallel scalability of these methods is demonstrated for some discretizations of infinite-dimensional variational inequalities.

Abstract: We consider the stochastic nonlinear complementarity problem (SNCP). We first formulate the problem as a stochastic mathematical program with equilibrium constraints and then, in order to develop efficient algorithms, we give some reformulations of the problem. Furthermore, based on the reformulations, we propose a smoothed penalty method for solving SNCP. A rigorous convergence analysis is also given.

Abstract: We consider the problem of discriminating between two finite point sets A and B in the n-dimensional space by using a special type of polyhedral function. An effective finite algorithm for finding a separating function based on iterative solutions of linear programming subproblems is suggested. At each iteration a function whose graph is a polyhedral cone with vertex at a certain point is constructed and the resulting separating function is defined as a point-wise minimum of these functions. It has been shown that arbitrary two finite point disjoint sets can be separated by using this algorithm. An illustrative example is given and an application on classification problems with some real-world data sets has been implemented.

Abstract: Cutting plane methods provide the means to solve large scale semidefinite programs (SDP) cheaply and quickly. They can also conceivably be employed for the purposes of re-optimization after branching or the addition of cutting planes. We give a survey of various cutting plane approaches for SDP in this paper. These cutting plane approaches arise from various perspectives, and include techniques based on interior point cutting plane approaches, non-differentiable optimization, and finally an approach which mimics the simplex method for linear programming (LP). We present an accessible introduction to various cutting plane approaches that have appeared in the literature. We place these methods in a unifying framework which illustrates how each approach arises as a natural enhancement of a primordial LP cutting plane scheme based on a semi-infinite formulation of the SDP.

Abstract: The quadratic assignment problem (QAP) is one of the great challenges in combinatorial optimization. Linearization for QAP is to transform the quadratic objective function into a linear one. Numerous QAP linearizations have been proposed, most of which yield mixed integer linear programs. Kauffmann and Broeckx’s linearization (KBL) is the current smallest one in terms of the number of variables and constraints. In this article, we give a new linearization, which has the same size as KBL. Our linearization is more efficient in terms of the tightness of the continuous relaxation. Furthermore, the continuous relaxation of our linearization leads to an improvement to the Gilmore–Lawler bound. We also give a corresponding cutting plane heuristic method for QAP and demonstrate its superiority by numerical results.

Abstract: Multi-class classification is an important and on-going research subject in machine learning. In this article, we propose a new support vector algorithm, called ν-K-SVCR, for multi-class classification based on ν-support vector machine. ν-K-SVCR has parameters that enable us to control the numbers of support vectors and margin errors effectively, which is helpful in improving the accuracy of each classifier. We give some theoretical results concerning the significance of the parameters and show the robustness of classifiers. In addition, we have examined the proposed algorithm on several benchmark data sets and artificial data sets, and our preliminary experiments confirm our theoretical conclusions.

Abstract: Both theory and implementations in deterministic global optimization have advanced significantly in the past decade. Two schools of thought have developed: the first employs various bounding techniques without validation, while the second employs different techniques, in a way that always rigorously takes account of roundoff error (i.e. with validation). However, convex relaxations, until very recently used without validation, can be implemented efficiently in a validated context. Here, we empirically compare a validated implementation of a variant of convex relaxations (linear relaxations applied to each intermediate operation) with traditional techniques from validated global optimization (interval constraint propagation and interval Newton methods). Experimental results show that linear relaxations are of significant value in validated global optimization, although further exploration will probably lead to more effective inclusion of the technology.

Abstract: This paper addresses a topic of supply chain management from a new perspective: incorporating risk aversion in a supply contract model. A pay to delay capacity reservation contract is analyzed by using the concept of conditional value-at-risk (Cvar). First, we construct the manufacturer’s optimal ordering problem by using the dynamic programming approach. Then, we derive the manufacturer’s optimal ordering strategy and compare our results with those obtained by using expectation performance and mean-variance tradeoff. Finally, numerical results are shown to reveal the impact of risk aversion on the manufacturer’s optimal decisions. The analysis presented in this paper reveals advantages of using the Cvar approach over the mean-variance approach.

Abstract: This article presents a concise review of the scientific–technical computing system Maple and its application potentials in Operations Research. The primary emphasis is placed on non-linear optimization models that may be formulated using a broad range of components, including e.g. special functions, integrals, systems of differential equations, deterministic or stochastic simulation, external function calls, and other computational procedures. Such models may have a complicated structure, with possibly a non-convex feasible set and multiple, global and local, optima. We introduce the Global Optimization Toolbox to solve such models in Maple, and illustrate its usage by numerical examples.

Abstract: We generalize new criss-cross type algorithms for linear complementarity problems (LCPs) given with sufficient matrices. Most LCP solvers require a priori information about the input matrix. The sufficiency of a matrix is hard to be checked (no polynomial time method is known). Our algorithm is similar to Zhang's linear programming and Akkeles¸, Balogh and Illes's criss-cross type algorithm for LCP-QP problems. We modify our basic algorithm in such a way that it can start with any matrix M, without having any information about the properties of the matrix (sufficiency, bisymmetry, positive definiteness, etc.) in advance. Even in this case, our algorithm terminates with one of the following cases in a finite number of steps: it solves the LCP problem, it solves its dual problem or it gives a certificate that the input matrix is not sufficient, thus cycling can occur. Although our algorithm is more general than that of Akkeles¸, Balogh and Illes's, the finiteness proof has been simplified. Furthermore, the ...

Abstract: This present paper is concerned with second-order methods for a class of shape optimization problems. We employ a complete boundary integral representation of the shape Hessian which involves first- and second-order derivatives of the state and the adjoint state function, as well as normal derivatives of its local shape derivatives. We introduce a boundary integral formulation to compute these quantities. The derived boundary integral equations are solved efficiently by a wavelet Galerkin scheme. A numerical example validates that, in spite of the higher effort of the Newton method compared to first-order algorithms, we obtain more accurate solutions in less computational time.

Abstract: In this article, the problem of determining the shape of a thin wing for minimum drag has been examined. In order to determine the optimal shape, we have extended a measure theory-based method. Using an embedding procedure, the problem of finding the optimal shape is reduced to one consisting of minimizing a linear form over a set of positive measures. The resulting problem can be approximated by a finite dimensional linear programming problem. The solution of this problem is used to construct a nearly optimal shape. A numerical example is given to illustrate the method.

Abstract: The Klee-Minty cube is a well-known worst case example for which the simplex method takes an exponential number of iterations as the algorithm visits all the 2 n vertices of the n-dimensional cube. While such behavior is excluded by polynomial interior point methods, we show that, by adding an exponential number of redundant inequalities, the central path can be bent along the edges of the Klee-Minty cube. More precisely, for an arbitrarily small δ, the central path takes 2 n −2 turns as it passes through the δ-neighborhood of all the vertices of the Klee-Minty cube in the same order as the simplex method does.

Abstract: The pivot algorithm and interior point algorithm are two diverging and competitive types of algorithms for solving linear programming problems. One of the main disadvantages of pivot algorithms is that they suffer from problems especially stalling, arising from degeneracy. Interior point methods, although not affected by degeneracy, require a purification procedure to obtain an exact optimal solution. This paper derives a new general-purpose solution algorithm, called combined projected gradient algorithm with the proof that it converges successfully. The new algorithm combines pivot and interior point techniques naturally. It in fact not only reduces the possibility of difficulty arising from degeneracy, but also reaches a pair of exact primal and dual optimal solutions. Numerical results on a group of NETLIB problems are encouraging.

Abstract: A globalization strategy for multigrid schemes solving optimal control problems is presented. This approach searches for possible negative eigenvalues of the reduced Hessian considered at the coarsest grid of the multigrid process. If negative eigenvalues are detected, a globalization step in the direction of negative curvature is performed to escape undesired maxima or saddle points. It is shown that the multigrid solution step provides a descent update. Examples are given to illustrate and validate the present approach.

Abstract: By using a smoothing function, the linear complementarity problem (LCP) can be reformulated as a parameterized smooth equation. A Newton method with a projection-type testing procedure is proposed to solve this equation. We show that, for the LCP with a sufficient matrix, the iteration sequence generated by the proposed algorithm is bounded as long as the LCP has a solution. This assumption is weaker than the ones used in most existing smoothing algorithms. Moreover, we show that the proposed algorithm can find a maximally complementary solution to the LCP in a finite number of iterations.

Abstract: We consider a general equilibrium problem defined on a convex set, whose cost bifunction may be nonmonotone. We show that this problem can be solved by the inexact partial proximal point method. These results can be viewed as a generalization of the known convergence properties of the usual proximal point method.

Abstract: In many real-world applications, the cost associate with maintenance could be very significant since the maintenance work is subcontract or repairmen have to make a special trip from some central place to carry out the work. The focus of this study is to optimally coordinate the maintenance schedule of machines to save the maintenance cost incurred, which is named as the maintenance scheduling problem for a family of machines (MSPFM). Before presenting our solution approach, we first review Goyal and Kusy's [Goyal, S.K. and Kusy, M.I., 1985, Determining economic maintenance frequency for a family of machines. Journal of the Operational Research Society, 36(12), 1125–1128.] model and their heuristic for solving the MSPFM. In this study, we conduct full analysis on the mathematical model for the MSPFM. By utilizing our theoretical results, we propose a new search algorithm that solves the optimal solution for the MSPFM very efficiently. Using 36,000 instances randomly generated, we show that our new search ...

Abstract: The paper introduces an identification problem arising in modern regional hyperthermia, a cancer therapy aiming at heating the tumor by microwave radiation. The task is to identify the highly individual perfusion, which affects the resulting temperature distribution, from magnetic resonance measurements. The identification problem is formulated as an optimization problem. Existence of a solution and optimality conditions are analyzed. For the numerical solution, an SQP method is used. Sufficient conditions for the convergence of the method are derived. Finally, numerical examples on artificial as well as clinical data are presented.

Abstract: In this paper, we present knowledge-based support vector machine (SVM) classifiers using semidefinite linear programming. SVMs are an optimization-based solution method for large-scale data classification problems. Knowledge-based SVM classifiers, where prior knowledge is in the form of ellipsoidal constraints, result in a semidefinite linear programme with a set containment constraint. These problems are reformulated as standard semidefinite linear programming problems by the application of a dual characterization of the set containment under a mild regularity condition. The reformulated semidefinite linear programme is solved by the publicly available solvers. Computational results show that prior knowledge can often improve correctness of the classifier.

Abstract: We consider two related nonlinear integer programming problems arising in series–parallel reliability systems: the constrained redundancy problem and the cost minimization problem. We propose in this paper an efficient method for solving these two types of nonlinear integer programming problems. The proposed convergent Lagrangian method combines Lagrangian relaxation with a duality reduction technique. An outer approximation method is used to search for the optimal dual solution and to generate Lagrangian bounds of the primal problem. To reduce the duality gap, we derive a special partition scheme by exploiting the inherent monotonicity and separability of the problem. Furthermore, a special optimality criterion is adopted to improve feasible solutions and to fathom integer subboxes. Computational results show that the algorithm is capable of solving large-scale optimization problems in series–parallel reliability systems. Comparison numerical results with other existing methods are also reported.

Abstract: For a maximal monotone operator T on a Hilbert space H and a closed subspace A of H, we consider the problem of finding (x, y∈T(x)) satisfying x∈A and y∈A ⊥ An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn The resulting generalized equation can be solved by using the proximal point algorithm We consider instead the use of hybrid proximal methods Hybrid methods use enlargements of operators, close in spirit to the concept of ϵ-subdifferentials We characterize the enlargement of the partial inverse operator in terms of the enlargement of T itself We present a new algorithm of resolution that combines Spingarn and hybrid methods, we prove for this method global convergence only assuming existence of solutions and maximal monotonicity of T We also show that, under standard assumptions, the method has a linear rate of convergence For the important problem of finding a zero of a sum of maximal monotone operators T 1, …, T m , we present a highly paralleliza

Abstract: In this paper, we study the distributed and boundary optimal control of the Stokes equations in the presence of pointwise control constraints. The analysis of the problems involve existence results, as well as necessary conditions and optimality systems. A primal–dual active set strategy is studied for the numerical solution of the problems. For the distributed case, global and local convergence results for the infinite dimensional method are presented. For the boundary control case, due to the lack of regularity of the pointwise constraint multiplier, the method is applied to a discretized version of the original problem. Finally, some numerical experiments, which illustrate the performance of the method for both cases, are commented.

Abstract: In this paper, we study some well-known cases of nonlinear programming problems, presenting them as instances of inexact or semi-infinite linear programming. The class of problems considered contains, in particular, semi-definite programming, second-order cone programming and special cases of inexact semi-definite programming. Strong duality results for the nonlinear problems studied are obtained via the Lagrangian duality. Using these results, we propose some dual algorithms for the studied classes of problems. The proposed algorithms can be interpreted as cutting plane or discretization algorithms. Finally, some comments on the convergence of the proposed algorithms and on preliminary numerical tests are given.

Abstract: We present a new definition of filled function, named globally concavized filled function, for the box constrained continuous global minimization problem. A class of this kind of filled functions are constructed. These functions contain two easily determinable parameters, which are not dependent on the radius of the basin at the current local minimizer to make them be globally concavized filled functions. We design a randomized algorithm to solve the problem basing on these functions and prove that the algorithm can converge asymptotically with probability one to a global minimizer of the problem. Numerical experiments are presented to show the effectiveness and robustness of the algorithm.

Abstract: For the validation of aerodynamic models, certain flight trajectories have to be chosen. In the current paper, the methodology of optimum experimental design is employed for the solution of this task. Aerodynamic models are highly nonlinear and so far no optimum experimental design approaches have been applied to them. We show that the methods of optimum experimental design are efficiently applicable to such models and considerably improve the statistical accuracy of the parameter estimates.

Abstract: Iterative learning control (ILC) and repetitive control (RC) use iterations in hardware that adjust the input to a system in order to converge to zero tracking error following a desired system output. ILC experiments on a robot improved the tracking accuracy during a high-speed manoeuvre by a factor of 1000 in approximately 12 iterations. Such performance requires knowing system phase information accurate to within ±90° or better. Otherwise, the iterations appear to start diverging. During divergence, they produce inputs that particularly excite unmodelled or poorly modelled dynamics, producing experimental data that is focussed on what is wrong with the current model. This article investigates use of RC/ILC for the purpose of developing good data sets for identification. This reverses the normal objective in RC/ILC to make convergence to zero tracking error as robust to model error as possible. Instead, for identification, one aims to make the convergence of iterations as sensitive as possible to model e...

Abstract: We describe an implementation of an infinite-dimensional primal–dual algorithm based on the Nesterov–Todd direction. Several applications to both continuous and discrete-time multi-criteria linear-quadratic control problems and linear-quadratic control problem with quadratic constraints are described. Numerical results show a very fast convergence (typically, within 3–4 iterations) to optimal solutions.

Abstract: Consider the class of nonlinear time-delayed optimal control problems with continuous linear constraints. This class of problems is difficult to solve numerically. In this article, a computational method based on a semi-infinite programming approach is given. This can be done by considering the control as the decision variable, while taking the state as a function of the control. After parametrizing the control, an approximated semi-infinite nonlinear problem is obtained. To solve this approximate problem, we use the dual parametrization method. The dual problem is a max–min problem, which can be solved by optimizing over two levels. The upper level can be solved by simulated annealing and the lower level can be solved by using any unconstrained optimisation technique, such as the quasi-Newton method.