Stochastic Multiobjective Optimization: Sample Average Approximation and Applications

We investigate one stage stochastic multiobjective optimization problems where the objectives are the expected values of random functions. Assuming that the closed form of the expected values is difficult to obtain, we apply the well known Sample Average Approximation (SAA) method to solve it. We propose a smoothing infinity norm scalarization approach to solve the SAA problem and analyse the convergence of efficient solution of the SAA problem to the original problem as sample sizes increase. Under some moderate conditions, we show that, with probability approaching one exponentially fast with the increase of sample size, an ϵ-optimal solution to the SAA problem becomes an ϵ-optimal solution to its true counterpart. Moreover, under second order growth conditions, we show that an efficient point of the smoothed problem approximates an efficient solution of the true problem at a linear rate. Finally, we describe some numerical experiments on some stochastic multiobjective optimization problems and report preliminary results.     

Vector coding algorithms for multidimensional discrete Fourier transform

A new fast algorithm is presented for the multidimensional discrete Fourier transform (DFT). This algorithm is derived using an interesting technique called “vector coding” (VC), and we call it the vector-coding fast Fourier transform (VC-FFT) algorithm. Since the VC-FFT is an extension of the Cooley–Tukey algorithm from 1-D to multidimensional form, the structure of the program is as simple as the Cooley–Tukey fast Fourier transform (FFT). The new algorithm significantly reduces the number of multiplications and recursive stages. The VC-FFT therefore comprehensively reduces the complexity of the algorithm as compared with other current multidimensional DFT algorithms.     

Accelerating Branch-and-Bound through a Modeling Language Construct for Relaxation-Specific Constraints

In the tradition of modeling languages for optimization, a single model is passed to a solver for solution. In this paper, we extend BARON’s modeling language in order to facilitate the communication of problem-specific relaxation information from the modeler to the branch-and-bound solver. This effectively results into two models being passed from the modeling language to the solver. Three important application areas are identified and computational experiments are presented. In all cases, nonlinear constraints are provided only to the relaxation constructor in order to strengthen the lower bounding step of the algorithm without complicating the local search process. In the first application area, nonlinear constraints from the reformulation–linearization technique (RLT) are added to strengthen a problem formulation. This approach is illustrated for the pooling problem and computational results show that it results in a scheme that makes global optimization nearly as fast as local optimization for pooling problems from the literature. In the second application area, we communicate with the relaxation constructor the first-order optimality conditions for unconstrained global optimization problems. Computational experiments with polynomial programs demonstrate that this approach leads to a significant reduction of the size of the branch-and-bound search tree. In the third application, problem-specific nonlinear optimality conditions for the satisfiability problem are used to strengthen the lower bounding step and are found to significantly expedite the branch-and-bound algorithm when applied to a nonlinear formulation of this problem.     

Levitin–Polyak Well-Posedness for Optimization Problems with Generalized Equilibrium Constraints

In this paper, we consider Levitin–Polyak well-posedness of parametric generalized equilibrium problems and optimization problems with generalized equilibrium constraints. Some criteria for these types of well-posedness are derived. In particular, under certain conditions, we show that generalized Levitin–Polyak well-posedness of a parametric generalized equilibrium problem is equivalent to the nonemptiness and compactness of its solution set. Finally, for an optimization problem with generalized equilibrium constraints, we also obtain that, under certain conditions, Levitin–Polyak well-posedness in the generalized sense is equivalent to the nonemptiness and compactness of its solution set.     

Numerical algorithm for a class of constrained optimal control problems of switched systems

In this paper, we consider an optimal control problem of switched systems with continuous-time inequality constraints. Because of the complexity of such constraints and switching laws, it is difficult to solve this problem by standard optimization techniques. To overcome the difficulty, we adopt a bi-level algorithm to divide the problem into two nonlinear constrained optimization problems: one continuous and the other discrete. To solve the problem, we transform the inequality constraints into equality constraints which is smoothed using a twice continuously differentiable function and treated as a penalty function. On this basis, the smoothed problem can be solved by any second-order gradient algorithm, e.g., Newton’s Method. Finally, numerical examples show that our method is effective compared to existing algorithms.     

Constrained Optimization: Projected Gradient Flows

We consider a dynamical system approach to solve finite-dimensional smooth optimization problems with a compact and connected feasible set. In fact, by the well-known technique of equalizing inequality constraints using quadratic slack variables, we transform a general optimization problem into an associated problem without inequality constraints in a higher-dimensional space. We compute the projected gradient for the latter problem and consider its projection on the feasible set in the original, lower-dimensional space. In this way, we obtain an ordinary differential equation in the original variables, which is specially adapted to treat inequality constraints (for the idea, see Jongen and Stein, Frontiers in Global Optimization, pp. 223–236, Kluwer Academic, Dordrecht, 2003). The article shows that the derived ordinary differential equation possesses the basic properties which make it appropriate to solve the underlying optimization problem: the longtime behavior of its trajectories becomes stationary, all singularities are critical points, and the stable singularities are exactly the local minima. Finally, we sketch two numerical methods based on our approach.     

An approximation algorithm for convex multi-objective programming problems

In multi-objective convex optimization it is necessary to compute an infinite set of nondominated points. We propose a method for approximating the nondominated set of a multi-objective nonlinear programming problem, where the objective functions and the feasible set are convex. This method is an extension of Benson’s outer approximation algorithm for multi-objective linear programming problems. We prove that this method provides a set of weakly ε-nondominated points. For the case that the objectives and constraints are differentiable, we describe an efficient way to carry out the main step of the algorithm, the construction of a hyperplane separating an exterior point from the feasible set in objective space. We provide examples that show that this cannot always be done in the same way in the case of non-differentiable objectives or constraints.     

A fast heuristic method for minimizing traffic congestion on reconfigurable ring topologies

We describe the development of fast heuristics and methodologies for congestion minimization problems in directional wireless networks, and we compare their performance with optimal solutions. The focus is on the network layer topology control problem (NLTCP) defined by selecting an optimal ring topology as well as the flows on it. Solutions to NLTCP need to be computed in near realtime due to changing weather and other transient conditions and which generally preclude traditional optimization strategies. Using a mixed-integer linear programming formulation, we present both new constraints for this problem and fast heuristics to solve it. The new constraints are used to increase the lower bound from the linear programming relaxation and hence speed up the solution of the optimization problem by branch and bound. The upper and lower bounds for the optimal objective function to the mixed integer problem then serve to evaluate new node-swapping heuristics which we also present. Through a series of tests on different sized networks with different traffic demands, we show that our new heuristics achieve within about 0.5% of the optimal value within seconds.     

Affine Systems: Asymptotics at Infinity for Fractal Measures

We study measures on ℝ ^d which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures μ induced by a finite family of affine mappings in ℝ ^d (the focus of our paper), as well as equilibrium measures in complex dynamics. By a systematic analysis of the Fourier transform of the measure μ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when μ is singular carry a gradation, ranging from Cantor-like fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of μ for paths at infinity in ℝ ^d . We show how properties of μ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in ℝ ^d , in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix.      

A fast algorithm for solving the dirichlet problem on hexagonal templet for the poisson equation in a rectangle

Using an explicit form of eigenvalues of the Laplacian on a hexagonal molecule, an economical method based on a fast Fourier transform is constructed for solving the Dirichlet problem in a rectangle. Bibliography:8 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 19–26     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.     

An augmented Lagrangian method for optimization problems with structured geometric constraints

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

     

一种求解修正的Helmholtz方程Cauchy问题的数值方法

本文研究了无界带形区域Ω={(x,y)|0相似文献   

Decomposition through formalization in a product space

When an optimization problem is posed in a product space it is classical to decompose this problem. The goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space. By associating systematically to this problem an equivalent one posed in ann-fold cartesian product space, we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former. Applications to three rather classical mathematical programming problems are given.     

Legendre Transform and Applications to Finite and Infinite Optimization

We investigate convex constrained nonlinear optimization problems and optimal control with convex state constraints in the light of the so-called Legendre transform. We use this change of coordinate to propose a gradient-like algorithm for mathematical programs, which can be seen as a search method along geodesics. We also use the Legendre transform to study the value function of a state constrained Mayer problem and we show that it can be characterized as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.     

<Emphasis Type="Italic">α</Emphasis>-Conservative approximation for probabilistically constrained convex programs

In this paper, we address an approximate solution of a probabilistically constrained convex program (PCCP), where a convex objective function is minimized over solutions satisfying, with a given probability, convex constraints that are parameterized by random variables. In order to approach to a solution, we set forth a conservative approximation problem by introducing a parameter α which indicates an approximate accuracy, and formulate it as a D.C. optimization problem.     

Convergence Analysis of a Class of Penalty Methods for Vector Optimization Problems with Cone Constraints

In this paper, we consider convergence properties of a class of penalization methods for a general vector optimization problem with cone constraints in infinite dimensional spaces. Under certain assumptions, we show that any efficient point of the cone constrained vector optimization problem can be approached by a sequence of efficient points of the penalty problems. We also show, on the other hand, that any limit point of a sequence of approximate efficient solutions to the penalty problems is a weekly efficient solution of the original cone constrained vector optimization problem. Finally, when the constrained space is of finite dimension, we show that any limit point of a sequence of stationary points of the penalty problems is a KKT stationary point of the original cone constrained vector optimization problem if Mangasarian–Fromovitz constraint qualification holds at the limit point.This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University.     

Multiobjective evolutionary algorithms for complex portfolio optimization problems

This paper investigates the ability of Multiobjective Evolutionary Algorithms (MOEAs), namely the Non-dominated Sorting Genetic Algorithm II (NSGA-II), Pareto Envelope-based Selection Algorithm (PESA) and Strength Pareto Evolutionary Algorithm 2 (SPEA2), for solving complex portfolio optimization problems. The portfolio optimization problem is a typical bi-objective optimization problem with objectives the reward that should be maximized and the risk that should be minimized. While reward is commonly measured by the portfolio’s expected return, various risk measures have been proposed that try to better reflect a portfolio’s riskiness or to simplify the problem to be solved with exact optimization techniques efficiently. However, some risk measures generate additional complexities, since they are non-convex, non-differentiable functions. In addition, constraints imposed by the practitioners introduce further difficulties since they transform the search space into a non-convex region. The results show that MOEAs, in general, are efficient and reliable strategies for this kind of problems, and their performance is independent of the risk function used.     

A feasible method for optimization with orthogonality constraints

Minimization with orthogonality constraints (e.g., $X^\top X = I$ ) and/or spherical constraints (e.g., $\Vert x\Vert _2 = 1$ ) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform—a Crank-Nicolson-like update scheme—to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 % to the best known solution on the largest problem “tai256c” in QAPLIB can be reached in 5 min on a typical laptop.     

Complexity and algorithms for nonlinear optimization problems

Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems providing optimal solutions with prespecified accuracy in the solution space. We delineate the complexity status of convex problems over network constraints, dual of flow constraints, dual of multi-commodity, constraints defined by a submodular rank function (a generalized allocation problem), tree networks, diagonal dominant matrices, and nonlinear knapsack problem’s constraint. All these problems, except for the latter in integers, have polynomial time algorithms which may be viewed within a unifying framework of a proximity-scaling technique or a threshold technique. The complexity of many of these algorithms is furthermore best possible in that it matches lower bounds on the complexity of the respective problems. In general nonseparable optimization problems are shown to be considerably more difficult than separable problems. We compare the complexity of continuous versus discrete nonlinear problems and list some major open problems in the area of nonlinear optimization. An earlier version of this paper appeared in 4OR, 3:3, 171–216, 2005.