Pure adaptive search in monte carlo optimization

Pure adaptive search constructs a sequence of points uniformly distributed within a corresponding sequence of nested regions of the feasible space. At any stage, the next point in the sequence is chosen uniformly distributed over the region of feasible space containing all points that are equal or superior in value to the previous points in the sequence. We show that for convex programs the number of iterations required to achieve a given accuracy of solution increases at most linearly in the dimension of the problem. This compares to exponential growth in iterations required for pure random search.     

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??We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász's Theta Function

We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász's theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its induced subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warm-start strategy to recompute the theta function on smaller subgraphs. Computational results demonstrate that our algorithm can efficiently extract maximum stable sets in comparable time it takes to solve the theta problem on the original graph to optimality. This work was supported in part by NSF through CAREER Grant DMI-0237415. Part of this work was performed while the first author was at the Department of Applied Mathematics and Statisticsat Stony Brook University, Stony Brook, NY, USA.     

Generalized fractional programming duality: A parametric approach

Using a parametric approach, duality is presented for a minimax fractional programming problem that involves several ratios in the objective function.The first author is thankful to Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319, and the authors are thankful to the anonymous referees for useful suggestions.     

A locally computed suboptimal control strategy for a class of interconnected systems

In this paper, a locally computed suboptimal control strategy for a class of interconnected systems is introduced. First, optimal statefeedback control equations are derived for a finite-horizon quadratic cost. Then, the control for each subsystem is separated into two portions. The first portion stabilizes the isolated subsystem, and the second portion corresponds to the interactions. To achieve a locally calculable control, an approximation to the optimal control equations is introduced, and two iterative suboptimal control algorithms are developed. In the first algorithm, the initial conditions of subsystems are assumed to be known; in the second algorithm, this information is replaced by statistical distributions. The orders of errors in the iterations of the algorithm and in the suboptimality are given in terms of interconnections. An example with comparisons is also included to show the performance of the approach.     

简并光学参量振荡器混沌反控制

提出一种实现简并光学参量振荡器混沌反控制的方法，用正弦信号调制简并光学参量振荡器的基模衰减率，使简并光学参量振荡器从定态输出转化为混沌态.数值模拟结果表明，选择不同的调制幅度和调制角频率，只要满足系统的最大李雅谱诺夫指数大于零，即可实现不同的混沌轨道重构.通过比较最大李雅谱诺夫指数λ_max随调制幅度和调制角频率变化曲线， 指出系统从周期态调制到混沌态比从无亚谐波输出的定态调制到混沌态更容易，有更宽的调制幅度和调制角频率选择范围. 关键词： 简并光学参量振荡器 混沌反控制 调制     

A finite cutting plane method for facial disjunctive programs

This paper shows that any linear disjunctive program with a finite number of constraints can be transformed into an equivalent facial program. Based upon linear programming technique, a new, finite cutting plane method is presented for the facial programs.

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Zusammenfassung Die Arbeit zeigt, daß jedes lineare disjunktive Optimierungsproblem mit endlich vielen Restriktionen in ein äquivalentes Fazetten-Problem transformiert werden kann. Auf der Grundlage von linearer Optimierungstechnik wird für das Fazetten-Problem ein neues, endliches Schnittebenenverfahren vorgestellt.

     

A new trust region technique for the maximum weight clique problem

A new simple generalization of the Motzkin-Straus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a Motzkin-Straus optimum coincides with such a point. The developed method has complexity O(n³), where n is the number of vertices of the graph. It was implemented in a publicly available software package QUALEX-MS.Computational experiments indicate that the algorithm is exact on small graphs and very efficient on the DIMACS benchmark graphs and various random maximum weight clique problem instances.     

变量有广义界线性规划的直接对偶单纯形法

本文讨论变量有广义界线性规划问题借助标准形线性规划同单纯形法技术，建立问题的一个直接对偶单纯形法。分析了方法的性质，给出了初始对偶可行基的计算方法，并用实例说明方法的具体操作。     

Redefining Event Variables for Efficient Modeling of Continuous-Time Batch Processing

We define events so as to reduce the number of events and decision variables needed for modeling batch-scheduling problems such as described in [15]. We propose a new MILP formulation based on this concept, defining non-uniform time periods as needed and decision variables that are not time-indexed. It can handle complicated multi-product/multi-stage machine processes, with production lines merging and diverging, and with minimum and maximum batch sizes. We compare it with earlier models and show that it can solve problems with small to medium demands relative to batch sizes in reasonable computer times.