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.     

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．     

一个解凸二次规划的预测-校正光滑化方法

本文为凸二次规划问题提出一个光滑型方法,它是Engelke和Kanzow提出的解线性规划的光滑化算法的推广。其主要思想是将二次规划的最优性K-T条件写成一个非线性非光滑方程组,并利用Newton型方法来解其光滑近似。本文的方法是预测-校正方法。在较弱的条件下,证明了算法的全局收敛性和超线性收敛性。     

Ф凹（-φ）凸混合单调算子不动点存在惟一性及其应用

该文引入了Ф凹（-φ）凸算子，统一处理了一类具有某种凹凸性的混合单调算子，在非紧非连续的条件下，利用单调叠代技巧证明了不动点的存在惟一，进而得到了具有α凹-凸、凹-（-α）凸、α凹-Guo凸、凹-Guo凸、e凹-Guo凸、e凹-凸、e凹-（-α）凸以及α1凹-（-α）凸等性质的混合单调算子的新不动点定理，并将所获结果应用于Hammerstein非线性积分方程．     

φ凹(-Ψ)凸混合单调算子不动点存在惟一性及其应用

该文引入了φ凹-(—ψ)凸算子,统一处理了一类具有某种凹凸性的混合单调算子,在非紧非连续的条件下,利用单凋叠代技巧证明了不动点的存在惟一,进而得到了具有α凹-凸、凹-(—α)凸、α凹-Guo凸、凹-Guo凸、e凹-Guo凸、e凹-凸、e凹-(—α)凸以及α_1凹-(—α_2)凸等性质的混合单调算子的新不动点定理,并将所获结果应用于Hammerstein非线性积分方程。     

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．     

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．     

一类具有连续变量的全局最优化问题的凸化方法

本文研究了一类非凸最优化问题的凸化方法与最优性条件的问题.利用构造含有参数的函数变换方法,将具有次正定性质的目标函数凸化,并获得了这一类非凸优化问题全局最优解的充要条件,推广了凸化方法在求解全局最优化问题方面的应用.     

低秩稀疏矩阵恢复的快速非单调交替极小化方法

针对低秩稀疏矩阵恢复问题的一个非凸优化模型,本文提出了一种快速非单调交替极小化方法.主要思想是对低秩矩阵部分采用交替极小化方法,对稀疏矩阵部分采用非单调线搜索技术来分别进行迭代更新.非单调线搜索技术是将单步下降放宽为多步下降,从而提高了计算效率.文中还给出了新算法的收敛性分析.最后,通过数值实验的比较表明,矩阵恢复的非单调交替极小化方法比原单调类方法更有效.     

求解无限维非线性互补问题的光滑化牛顿法

研究一类无限维非线性互补问题的光滑化牛顿法.借助于非线性互补函数,将无限维非线性互补问题转化为一个非光滑算子方程.构造光滑算子逼近非光滑算子,在光滑逼近算子满足方向可微相容性的条件下,证明了光滑化牛顿法具有超线性收敛性.     

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.     

MONOTONIZATION IN GLOBAL OPTIMIZATION

A general monotonization method is proposed for converting a constrained programming problem with non-monotone objective function and monotone constraint functions into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavefication methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.     

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.     

A Computational Study of the Homogeneous Algorithm for Large-scale Convex Optimization

Recently the authors have proposed a homogeneous and self-dual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interior-point type method; nevertheless, it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems, which also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for large-scale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient.     

Stable monotone variational inequalities

Variational inequalities associated with monotone operators (possibly nonlinear and multivalued) and convex sets (possibly unbounded) are studied in reflexive Banach spaces. A variety of results are given which relate to a stability concept involving a natural parameter. These include characterizations useful as criteria for stable existence of solutions and also several characterizations of surjectivity. The monotone complementarity problem is covered as a special case, and the results are sharpened for linear monotone complementarity and for generalized linear programming.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 at the University of Wisconsin - Madison and by the National Science Foundation under Grant No. DMS-8405179 at the University of Illinois at Urbana-Champaign.     

Saddle-Point Optimality: A Look Beyond Convexity

The fact that two disjoint convex sets can be separated by a plane has a tremendous impact on optimization theory and its applications. We begin the paper by illustrating this fact in convex and partly convex programming. Then we look beyond convexity and study general nonlinear programs with twice continuously differentiable functions. Using a parametric extension of the Liu-Floudas transformation, we show that every such program can be identified as a relatively simple structurally stable convex model. This means that one can study general nonlinear programs with twice continuously differentiable functions using only linear programming, convex programming, and the inter-relationship between the two. In particular, it follows that globally optimal solutions of such general programs are the limit points of optimal solutions of convex programs.     

A block monotone iterative method for numerical solutions of nonlinear elliptic boundary value problems

The aim of this article is to develop a new block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem. The boundary value problem is discretized into a system of nonlinear algebraic equations, and a block monotone iterative method is established for the system using an upper solution or a lower solution as the initial iteration. The sequence of iterations can be computed in a parallel fashion and converge monotonically to a maximal solution or a minimal solution of the system. Three theoretical comparison results are given for the sequences from the proposed method and the block Jacobi monotone iterative method. The comparison results show that the sequence from the proposed method converges faster than the corresponding sequence given by the block Jacobi monotone iterative method. A simple and easily verified condition is obtained to guarantee a geometric convergence of the block monotone iterations. The numerical results demonstrate advantages of this new approach. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011