Stochastic Nonstationary Optimization for Finding Universal Portfolios

We apply ideas from stochastic optimization for defining universal portfolios. Universal portfolios are that class of portfolios which are constructed directly from the available observations of the stocks behavior without any assumptions about their statistical properties. Cover [7] has shown that one can construct such portfolio using only observations of the past stock prices which generates the same asymptotic wealth growth as the best constant rebalanced portfolio which is constructed with the full knowledge of the future stock market behavior.In this paper we construct universal portfolios using a different set of ideas drawn from nonstationary stochastic optimization. Our portfolios yield the same asymptotic growth of wealth as the best constant rebalanced portfolio constructed with the perfect knowledge of the future and they are less demanding computationally compared to previously known universal portfolios. We also present computational evidence using New York Stock Exchange data which shows, among other things, superior performance of portfolios which explicitly take into account possible nonstationary market behavior.     

Stochastic Method for the Solution of Unconstrained Vector Optimization Problems

We propose a new stochastic algorithm for the solution of unconstrained vector optimization problems, which is based on a special class of stochastic differential equations. An efficient algorithm for the numerical solution of the stochastic differential equation is developed. Interesting properties of the algorithm enable the treatment of problems with a large number of variables. Numerical results are given.     

Nested Partitions Method for Stochastic Optimization

The nested partitions (NP) method is a recently proposed new alternative for global optimization. Primarily aimed at problems with large but finite feasible regions, the method employs a global sampling strategy that is continuously adapted via a partitioning of the feasible region. In this paper we adapt the original NP method to stochastic optimization where the performance is estimated using simulation. We prove asymptotic convergence of the new method and present a numerical example to illustrate its potential.     

Stochastic Particle Method for Nonlinear Hyperbolic Problems

In this paper, we present a new method based on stochastic particles, which allows us to compute solutions of a system of nonlinear transport equations arising in the modeling of immiscible displacement in porous pedia. In this approach, we use different particles for different phases and move them according to the stochastic rules for which the probability density function depends on the spatial distribution of the particles. Our motivation for such a method is a Lagrangian modeling framework in which one can describe certain physical phenomena more naturally than in an Eulerian framework. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic Enumeration Method for Counting NP-Hard Problems

We present a new generic sequential importance sampling algorithm, called stochastic enumeration (SE) for counting #P-complete problems, such as the number of satisfiability assignments and the number of perfect matchings (permanent). We show that SE presents a natural generalization of the classic one-step-look-ahead algorithm in the sense that it: Runs in parallel multiple trajectories instead of a single one; Employs a polynomial time decision making oracle, which can be viewed as an n-step-look-ahead algorithm, where n is the size of the problem. Our simulation studies indicate good performance of SE as compared with the well-known splitting and SampleSearch methods.     

Two-Stage Nested Partitions Method for Stochastic Optimization

We address the problem of optimizing over a large but finite set when the objective function does not have an analytical expression and is evaluated using noisy estimation. Building on the recently proposed nested partitions method for stochastic optimization, we develop a new approach that combines this random search method and statistical selection for guiding the search. We prove asymptotic convergence and analyze the finite time behavior of the new approach. We also report extensive numerical results to illustrate the benefits of the new approach.     

General Convergence Analysis of Stochastic First-Order Methods for Composite Optimization

Journal of Optimization Theory and Applications - In this paper, we consider stochastic composite convex optimization problems with the objective function satisfying a stochastic bounded gradient...     

Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems

We study two-stage, finite-scenario stochastic versions of several combinatorial optimization problems, and provide nearly tight approximation algorithms for them. Our problems range from the graph-theoretic (shortest path, vertex cover, facility location) to set-theoretic (set cover, bin packing), and contain representatives with different approximation ratios. The approximation ratio of the stochastic variant of a typical problem is found to be of the same order of magnitude as its deterministic counterpart. Furthermore, we show that common techniques for designing approximation algorithms such as LP rounding, the primal-dual method, and the greedy algorithm, can be adapted to obtain these results.     

Parallelizable Preprocessing Method for Multistage Stochastic Programming Problems

Stochastic programming has extensive applications in practical problems such as production planning and portfolio selection. Typically, the model has very large size and some techniques are often used to exploit the special structure of the programs. It has been noticed that the coefficient matrix may not be of full rank in the well-known scenario formulation of stochastic programming; thus, the preprocessing is often necessary in developing rapid decomposition methods. In this paper, we propose a parallelizable preprocessing method, which exploits effectively the structure of the formulation. Although the underlying idea is simple, the method turns out to be very useful in practice, since it may help us to select the nonanticipativity constraints efficiently. Some numerical results are reported confirming the usefulness of the method.This work was partially supported by the Informatics Research Center for Development of Knowledge Society Infrastructure, Graduate School of Informatics, Kyoto University, Kyoto, Japan. The work of the first author was also supported in part by the National Science Foundation of China, Grant 10571039. The work of the second author was also supported in part by the Scientific Research Grant-in-Aid from the Japan Society for the Promotion of Science. The authors are grateful to the referees for careful reading of the paper and helpful comments.This author’s work was done while he was visiting Kyoto University.     

Universal Method of Searching for Equilibria and Stochastic Equilibria in Transportation Networks

Computational Mathematics and Mathematical Physics - A universal method of searching for usual and stochastic equilibria in congestion population games is proposed. The Beckmann and stable dynamics...     

An Adaptive Hybrid Spectral Method for Stochastic Helmholtz Problems

The implementation of an adaptive hybrid spectral method for Helmholtz equations with random parameters is addressed in this work. New error indicators for generalized polynomial chaos for stochastic approximations and spectral element methods for physical approximations are developed, and systematic adaptive strategies are proposed associated with these error indicators. Numerical results show that these error indicators provide effective estimates for the approximation errors, and the overall adaptive procedure results in efficient approximation method for the stochastic Helmholtz equations.     

Expected Residual Minimization Method for Stochastic Variational Inequality Problems

This paper considers a stochastic variational inequality problem (SVIP). We first formulate SVIP as an optimization problem (ERM problem) that minimizes the expected residual of the so-called regularized gap function. Then, we focus on a SVIP subclass in which the function involved is assumed to be affine. We study the properties of the ERM problem and propose a quasi-Monte Carlo method for solving the problem. Comprehensive convergence analysis is included as well. This work was supported in part by SRF for ROCS, SEM and Project 10771025 supported by NSFC.     

随机线性互补问题的无约束优化再定式

针对随机线性互补问题,提出等价的无约束优化再定式模型,即由D-间隙函数定义的确定性的无约束期望残差极小化问题.通过拟Monte Carlo方法,将样本进行了推广,得到了相关的离散近似问题.在适当的条件下,提出了最优解存在的充分条件,以及探究了离散近似问题的最优解及稳定点的收敛性.另外,在针对一类带有常系数矩阵的随机互补线性问题,研究了解存在的充要条件.     

A Stochastic/Perturbation Global Optimization Algorithm for Distance Geometry Problems

We present a new global optimization approach for solving exactly or inexactly constrained distance geometry problems. Distance geometry problems are concerned with determining spatial structures from measurements of internal distances. They arise in the structural interpretation of nuclear magnetic resonance data and in the prediction of protein structure. These problems can be naturally formulated as global optimization problems which generally are large and difficult. The global optimization method that we present is related to our previous stochastic/perturbation global optimization methods for finding minimum energy configurations, but has several key differences that are important to its success. Our computational results show that the method readily solves a set of artificial problems introduced by Moré and Wu that have up to 343 atoms. On a set of considerably more difficult protein fragment problems introduced by Hendrickson, the method solves all the problems with up to 377 atoms exactly, and finds nearly exact solution for all the remaining problems which have up to 777 atoms. These preliminary results indicate that this approach has very good promise for helping to solve distance geometry problems.     

复合凸优化问题的Fenchel-Lagrange强对偶之研究

利用共轭函数的上图性质,引入新的约束规范条件,等价刻画了目标函数为凸函数与凸复合函数之和的复合优化问题及其Fenchel-Lagrange对偶问题之间的强对偶与稳定强对偶.     

A Level-Value Estimation Method and Stochastic Implementation for Global Optimization

In this paper, we propose a new method, namely the level-value estimation method, for finding global minimizer of continuous optimization problem. For this purpose, we define the variance function and the mean deviation function, both depend on a level value of the objective function to be minimized. These functions have some good properties when Newton’s method is used to solve a variance equation resulting by setting the variance function to zero. We prove that the largest root of the variance equation equals the global minimal value of the corresponding optimization problem. We also propose an implementable algorithm of the level-value estimation method where importance sampling is used to calculate integrals of the variance function and the mean deviation function. The main idea of the cross-entropy method is used to update the parameters of sample distribution at each iteration. The implementable level-value estimation method has been verified to satisfy the convergent conditions of the inexact Newton method for solving a single variable nonlinear equation. Thus, convergence is guaranteed. The numerical results indicate that the proposed method is applicable and efficient in solving global optimization problems.     

Stability Analysis for Composite Optimization Problems and Parametric Variational Systems

This paper aims to provide various applications for second-order variational analysis of extended-real-valued piecewise linear functions recently obtained by the authors. We mainly focus here on establishing relationships between full stability of local minimizers in composite optimization and Robinson’s strong regularity of associated (linearized and nonlinearized) KKT systems. Finally, we address Lipschitzian stability of parametric variational systems with convex piecewise linear potentials.