Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.     

Finding nonoscillatory solutions to difference schemes for the advection equation

The advection equation is solved using a weighted adaptive scheme that combines a monotone scheme with the central-difference approximation of the first spatial derivative. The determination of antidiffusion fluxes is treated as an optimization problem. The solvability of the optimization problem is analyzed, and the differential properties of the cost functional are examined. It is shown that the determination of antidiffusion fluxes is reduced to a linear programming problem in the case of an explicit scheme and to a nonlinear programming problem or a sequence of linear programming problems in the case of an implicit scheme. A simplified monotonization algorithm is proposed. Numerical results are presented.     

一些类型的数学规划问题的全局最优解

本文对严格单调函数给出了几个凸化和凹化的方法，利用这些方法可将一个严格单调的规划问题转化为一个等价的标准D．C．规划或凹极小问题．本文还对只有一个严格单调的约束的非单调规划问题给出了目标函数的一个凸化和凹化方法，利用这些方法可将只有一个严格单调约束的非单调规划问题转化为一个等价的凹极小问题．再利用已有的关于D．C．规划和凹极小的算法，可以求得原问题的全局最优解．     

A convexification method for a class of global optimization problems with applications to reliability optimization

A convexification method is proposed for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems can be transformed into equivalent concave minimization problems using the proposed convexification schemes. An outer approximation method can then be used to find the global solution of the transformed problem. Applications to mixed-integer nonlinear programming problems arising in reliability optimization of complex systems are discussed and satisfactory numerical results are presented.     

Convexification and Concavification for a General Class of Global Optimization Problems

A kind of general convexification and concavification methods is proposed for solving some classes of global optimization problems with certain monotone properties. It is shown that these minimization problems can be transformed into equivalent concave minimization problem or reverse convex programming problem or canonical D.C. programming problem by using the proposed convexification and concavification schemes. The existing algorithms then can be used to find the global solutions of the transformed problems.     

单调全局最优化问题的凸化外逼近算法

单调优化是指目标函数与约束函数均为单调函数的全局优化问题．本文提出一种新的凸化变换方法把单调函数化为凸函数，进而把单调优化问题化为等价的凸极大或凹极小问题，然后采用Hoffman的外逼近方法来求得问题的全局最优解．我们把这种凸化方法同Tuy的Polyblock外逼近方法作了比较，通过数值比较可以看出本文提出的凸化的方法在收敛速度上明显优于Polyblock方法．     

Monotone insertion and monotone extension of frame homomorphisms

The purpose of this paper is to introduce monotonization in the setting of pointfree topology. More specifically, monotonically normal locales are characterized in terms of monotone insertion and monotone extensions theorems.     

Monotone false discovery rate

This paper proposes a procedure to obtain monotone estimates of both the local and the tail false discovery rates that arise in large-scale multiple testing. The proposed monotonization is asymptotically optimal for controlling the false discovery rate and also has many attractive finite-sample properties.     

一些约束规划问题的近似全局最优解

首先将一个具有多个约束的规划问题转化为一个只有一个约束的规划问题,然后通过利用这个单约束的规划问题,对原来的多约束规划问题提出了一些凸化、凹化的方法,这样这些多约束的规划问题可以被转化为一些凹规划、反凸规划问题．最后,还证明了得到的凹规划和反凸规划的全局最优解就是原问题的近似全局最优解．     

Applications of the method of partial inverses to convex programming: Decomposition

A primal–dual decomposition method is presented to solve the separable convex programming problem. Convergence to a solution and Lagrange multiplier vector occurs from an arbitrary starting point. The method is equivalent to the proximal point algorithm applied to a certain maximal monotone multifunction. In the nonseparable case, it specializes to a known method, the proximal method of multipliers. Conditions are provided which guarantee linear convergence.This research was sponsored, in part, by the Air Force Office of Scientific Research under grant 80-0195.     

半定规划问题的一种新的预测-校正算法

本文首先将半定规划转化为一个变分不等式问题,在满足单调性和Lipschitz连续的条件下,提出了一种基于Korpelevich-Khobotv算法的新的预测-校正算法,并给出算法的收敛性分析.     

一种求解半定规划的邻近外梯度算法

本文提出了一种求解半定规划的邻近外梯度算法.通过转化半定规划的最优性条件为变分不等式,在变分不等式满足单调性和Lipschitz连续的前提下,构造包含原投影区域的半空间,产生邻近点序列来逼近变分不等式的解,简化了投影的求解过程.将该算法应用到教育测评问题中,数值实验结果表明,该方法是解大规模半定规划问题的一种可行方法.     

On difference of two monotone operators

In this paper, we introduce and consider the problem of finding zeroes of difference of two monotone operators in a Hilbert space. Using the resolvent operator technique, we show that this problem is equivalent to the fixed point problem. This equivalence is used to suggest and analyze an iterative method for finding a zero of difference of two monotone operators. We also discuss the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other techniques.     

A Douglas–Rachford splitting method for solving equilibrium problems

We propose a splitting method for solving equilibrium problems involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of the sum of two appropriate maximally monotone operators under a suitable qualification condition. Our algorithm is a consequence of the Douglas–Rachford splitting applied to this auxiliary monotone inclusion. Connections between monotone inclusions and equilibrium problems are studied.     

A path-following interior-point algorithm for linear and quadratic problems

We describe an algorithm for the monotone linear complementarity problem (LCP) that converges from any positive, not necessarily feasible, starting point and exhibits polynomial complexity if some additional assumptions are made on the starting point. If the problem has a strictly complementarity solution, the method converges subquadratically. We show that the algorithm and its convergence properties extend readily to the mixed monotone linear complementarity problem and, hence, to all the usual formulations of the linear programming and convex quadratic programming problems.This research was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

非线性规划的凸化,凹化和单调化

本文提出了一个指数型凸化,凹化变换,并证明了单调非线性规划总能变换成相应的凹规划或凸规划.还证明了带某种类型线性或非线性约束的非线性规划在适当条件下能变换成单调非线性规划.     

Extended Projection Methods for Monotone Variational Inequalities

In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities.     

一种大规模规划的分解——调协解法

将一个大的主问题分解成若干个小的子问题,一方面对子问题寻优,一方面逐步调整主问题与子问题之间、子问题与子问题之间的关系,最后达到主问题最优,这就是解大规模规划问题的主要手段之一——分解-调协法。本文将一个大规模规划问题分解成由若干个子规划组成的多目标规划的序列,并在后一问题的解集合序列上求出前一问题的最     

Modified barrier functions (theory and methods)

The nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and/or the objective function into an equivalent problem, the classical Lagrangian which has important properties on the primal and the dual spaces.The application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions (MBF's) and MBF Methods (MBFM's). Being classical Lagrangians (CL's) for an equivalent problem, the MBF's combine the best properties of the CL's and classical barrier functions (CBF's) but at the same time are free of their most essential deficiencies.Due to the excellent MBF properties, new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed.The MBFM have up to a superlinear rate of convergence and are to the classical barrier functions (CBF's) method as the Multipliers Method for Augmented Lagrangians is to the Classical Penalty Function Method. Based on the dual theory associated with MBF, the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed. The application of the MBF to linear (LP) and quadratic (QP) programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the CBF methods.The numerical realization of the MBFM leads to the Newton Modified Barrier Method (NMBM). The excellent MBF properties allow us to discover that for any nondegenerate constrained optimization problem, there exists a hot start, from which the NMBM has a better rate of convergence, a better complexity bound, and is more stable than the interior point methods, which are based on the classical barrier functions.     

交替方向法求解带线性约束的变分不等式

１引言变分不等式是一个有广泛应用的数学问题,它的一般形式是：确定一个向量,使其满足这里ｆ是一个从到自身的一个映射,Ｓ是Ｒ中的一个闭凸集．在许多实际问题中集合Ｓ往往具有如下结构其中ＡｂＫ是中的一个简单闭凸集．例如一个正卦限,一个框形约束结构,或者一个球简言之,Ｓ是Ｒ中的一个超平面与一个简单闭凸集的交．求解问题（１）－（２）,往往是通过对线性约束Ａ引人Ｌａｇｒａｎｇｅ乘子,将原问题化为如下的变分不等式：确定使得我们记问题（３）－（４）为ＶＩ（Ｆ）．熟知［３］,ＶＩ（,Ｆ）等价于投影方程其中凡（·）表…