线性不等式约束的广义非线性互补问题的仿射内点信赖域方法

提供了一种新的非单调内点回代线搜索技术的仿射内点信赖域方法解线性不等式约束的广义非线性互补问题(GCP).基于广义互补问题构成的半光滑方程组的广义Jacobian矩阵,算法使用l_2范数作为半光滑方程组的势函数,形成的信赖域子问题为一个带椭球约束的线性化的二次模型.利用广义牛顿方程计算试探迭代步,通过内点映射回代技术确保迭代点是严格内点,保证了算法的整体收敛性.在合理的条件下,证明了信赖域算法在接近最优点时可转化为广义拟牛顿步,进而具有局部超线性收敛速率.非单调技术将克服高度非线性情况加速收敛进展.最后,数值结果表明了算法的有效性.     

Approximate Subgradient Methods for Nonlinearly Constrained Network Flow Problems

The minimization of nonlinearly constrained network flow problems can be performed by using approximate subgradient methods. The idea is to solve this kind of problem by means of primal-dual methods, given that the minimization of nonlinear network flow problems can be done efficiently exploiting the network structure. In this work, it is proposed to solve the dual problem by using ε-subgradient methods, as the dual function is estimated by minimizing approximately a Lagrangian function, which includes the side constraints (nonnetwork constraints) and is subject only to the network constraints. Some well-known subgradient methods are modified in order to be used as ε-subgradient methods and the convergence properties of these new methods are analyzed. Numerical results appear very promising and effective for this kind of problems This research was partially supported by Grant MCYT DPI 2002-03330.     

Superlinearly convergent affine scaling interior trust-region method for linear constrained LC<Superscript>1</Superscript> minimization

We extend the classical affine scaling interior trust region algorithm for the linear constrained smooth minimization problem to the nonsmooth case where the gradient of objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear constrained LC1 optimization where the second-order derivative of the objective function is explicitly required to be locally Lipschitzian. The general trust region subproblem in the proposed algorithm is defined by minimizing an augmented affine scaling quadratic model which requires both first and second order information of the objective function subject only to an affine scaling ellipsoidal constraint in a null subspace of the augmented equality constraints. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. Applications of the algorithm to some nonsmooth optimization problems are discussed.     

Dual multiplicative algorithms for an entropy-linear programming problem

A multiplicative-barrier generalization of the Cauchy gradient descent method is proposed and studied. The technique is used to search for dual variables in the entropy maximization problem with affine constraints, which arises, for example, in the simulation of equilibria in macroscopic systems. For this class of problems, the dual variables can be used to effectively determine the primal ones. The global convergence of the iterative algorithms proposed is proved.     

An alternating direction method for linear‐constrained matrix nuclear norm minimization

The aim of the nuclear norm minimization problem is to find a matrix that minimizes the sum of its singular values and satisfies some constraints simultaneously. Such a problem has received more attention largely because it is closely related to the affine rank minimization problem, which appears in many control applications including controller design, realization theory, and model reduction. In this paper, we first propose an exact version alternating direction method for solving the nuclear norm minimization problem with linear equality constraints. At each iteration, the method involves a singular value thresholding and linear matrix equations which are solved exactly. Convergence of the proposed algorithm is followed directly. To broaden the capacity of solving larger problems, we solve approximately the subproblem by an iterative method with the Barzilai–Borwein steplength. Some extensions to the noisy problems and nuclear norm regularized least‐square problems are also discussed. Numerical experiments and comparisons with the state‐of‐the‐art method FPCA show that the proposed method is effective and promising. Copyright © 2011 John Wiley & Sons, Ltd.     

求解含平衡约束数学规划的熵函数法

提出求解含平衡约束数学规划问题(简记为MPEC问题)的熵函数法，在将原问题等价改写为单层非光滑优化问题的基础上，通过熵函数逼近，给出求解MPEC问题的序列光滑优化方法，证明了熵函数逼近问题解的存在性和算法的全局收敛性，数值算例表明了算法的有效性。     

A new variable reduction technique for convex integer quadratic programs

Convex integer quadratic programming involves minimization of a convex quadratic objective function with affine constraints and is a well-known NP-hard problem with a wide range of applications. We proposed a new variable reduction technique for convex integer quadratic programs (IQP). Based on the optimal values to the continuous relaxation of IQP and a feasible solution to IQP, the proposed technique can be applied to fix some decision variables of an IQP simultaneously at zero without sacrificing optimality. Using this technique, computational effort needed to solve IQP can be greatly reduced. Since a general convex bounded IQP (BIQP) can be transformed to a convex IQP, the proposed technique is also applicable for the convex BIQP. We report a computational study to demonstrate the efficacy of the proposed technique in solving quadratic knapsack problems.     

On certain optimization methods with finite-step inner algorithms for convex finite-dimensional problems with inequality constraints

Numerical methods are proposed for solving finite-dimensional convex problems with inequality constraints satisfying the Slater condition. A method based on solving the dual to the original regularized problem is proposed and justified for problems having a strictly uniformly convex sum of the objective function and the constraint functions. Conditions for the convergence of this method are derived, and convergence rate estimates are obtained for convergence with respect to the functional, convergence with respect to the argument to the set of optimizers, and convergence to the g-normal solution. For more general convex finite-dimensional minimization problems with inequality constraints, two methods with finite-step inner algorithms are proposed. The methods are based on the projected gradient and conditional gradient algorithms. The paper is focused on finite-dimensional problems obtained by approximating infinite-dimensional problems, in particular, optimal control problems for systems with lumped or distributed parameters.     

A trust region and affine scaling interior point method for nonconvex minimization with linear inequality constraints

A trust region and affine scaling interior point method (TRAM) is proposed for a general nonlinear minimization with linear inequality constraints [8]. In the proposed approach, a Newton step is derived from the complementarity conditions. Based on this Newton step, a trust region subproblem is formed, and the original objective function is monotonically decreased. Explicit sufficient decrease conditions are proposed for satisfying the first order and second order necessary conditions.?The objective of this paper is to establish global and local convergence properties of the proposed trust region and affine scaling interior point method. It is shown that the proposed explicit decrease conditions are sufficient for satisfy complementarity, dual feasibility and second order necessary conditions respectively. It is also established that a trust region solution is asymptotically in the interior of the proposed trust region subproblem and a properly damped trust region step can achieve quadratic convergence. Received: January 29, 1999 / Accepted: November 22, 1999?Published online February 23, 2000     

Convergent Outer Approximation Algorithms for Solving Unary Programs

Interesting cutting plane approaches for solving certain difficult multiextremal global optimization problems can fail to converge. Examples include the concavity cut method for concave minimization and Ramana's recent outer approximation method for unary programs which are linear programming problems with an additional constraint requiring that an affine mapping becomes unary. For the latter problem class, new convergent outer approximation algorithms are proposed which are based on sufficiently deep l-norm or quadratic cuts. Implementable versions construct optimal simplicial inner approximations of Euclidean balls and of intersections of Euclidean balls with halfspaces, which are of general interest in computational convexity. Computational behavior of the algorithms depends crucially on the matrices involved in the unary condition. Potential applications to the global minimization of indefinite quadratic functions subject to indefinite quadratic constraints are shown to be practical only for very small problem sizes.     

Sign reversion approach to concave minimization problems

We consider a problem of minimization of a concave function subject to affine constraints. By using sign reversion techniques we show that the initial problem can be transformed into a family of concave maximization problems. This property enables us to suggest certain algorithms based on the parametric dual optimization problem.     

线性不等式约束的广义非线性互补问题的仿射内点信赖域方法

提供了一种新的非单调内点回代线搜索技术的仿射内点信赖域方法解线性不等式约束的广义非线性互补问题(GCP).基于广义互补问题构成的半光滑方程组的广义Jacobian矩阵,算法使用l2范数作为半光滑方程组的势函数,形成的信赖域子问题为一个带椭球约束的线性化的二次模型.利用广义牛顿方程计算试探迭代步,通过内点映射回代技术确保迭代点是严格内点,保证了算法的整体收敛性.在合理的条件下,证明了信赖域算法在接近最优点时可转化为广义拟牛顿步,进而具有局部超线性收敛速率.非单调技术将克服高度非线性情况加速收敛进展.最后,数值结果表明了算法的有效性.     

Heuristic Methods for Linear Multiplicative Programming

The linear multiplicative programming is the minimization of the product of affine functions over a polyhedral set. The problem with two affine functions reduces to a parametric linear program and can be solved efficiently. For the objective function with more than two affine functions multiplied, no efficient algorithms that solve the problem to optimality have been proposed, however Benson and Boger have proposed a heuristic algorithm that exploits links of the problem with concave minimization and multicriteria optimization. We will propose a heuristic method for the problem as well as its modification to enhance the accuracy of approximation. Computational experiments demonstrate that the method and its modification solve randomly generated problems within a few percent of relative error.     

Constrained derivative-free optimization on thin domains

Many derivative-free methods for constrained problems are not efficient for minimizing functions on “thin” domains. Other algorithms, like those based on Augmented Lagrangians, deal with thin constraints using penalty-like strategies. When the constraints are computationally inexpensive but highly nonlinear, these methods spend many potentially expensive objective function evaluations motivated by the difficulties in improving feasibility. An algorithm that handles this case efficiently is proposed in this paper. The main iteration is split into two steps: restoration and minimization. In the restoration step, the aim is to decrease infeasibility without evaluating the objective function. In the minimization step, the objective function f is minimized on a relaxed feasible set. A global minimization result will be proved and computational experiments showing the advantages of this approach will be presented.     

Spectral projected gradient and variable metric methods for optimization with linear inequalities

A family of variable metric methods for convex constrained optimizationwas introduced recently by Birgin, Martínez and Raydan.One of the members of this family is the inexact spectral projectedgradient (ISPG) method for minimization with convex constraints.At each iteration of these methods a strictly convex quadraticfunction with convex constraints must be (inexactly) minimized.In the case of the ISPG method it was shown that, in some importantapplications, iterative projection methods can be used for thisminimization. In this paper the particular case in which theconvex domain is a polytope described by a finite set of linearinequalities is considered. For solving the linearly constrainedconvex quadratic subproblem a dual approach is adopted, by meansof which subproblems become (not necessarily strictly) convexquadratic minimization problems with box constraints. Thesesubproblems are solved by means of an active-set box-constraintquadratic optimizer with a proximal-point type unconstrainedalgorithm for minimization within the current faces. Convergenceresults and numerical experiments are presented.     

Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.     

An implementable proximal point algorithmic framework for nuclear norm minimization

The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature.     

Nonlinear programming methods in the presence of noise

The problem of minimizing a nonlinear function with nonlinear constraints when the values of the objective, the constraints and their gradients have errors, is studied. This noise may be due to the stochastic nature of the problem or to numerical error.Various previously proposed methods are reviewed. Generally, the minimization algorithms involve methods of subgradient optimization, with the constraints introduced through penalty, Lagrange, or extended Lagrange functions. Probabilistic convergence theorems are obtained. Finally, an algorithm to solve the general convex (nondifferentiable) programming problem with noise is proposed.Originally written for presentation at the 1976 Budapest Symposium on Mathematical Programming.     

Branch-and-bound decomposition approach for solving quasiconvex-concave programs

A class of branch-and-bound methods is proposed for minimizing a quasiconvex-concave function subject to convex and quasiconvex-concave inequality constraints. Several important special cases where the subproblems involved by the bounding-and-branching operations can be solved quite effectively include certain d.c. programming problems, indefinite quadratic programming with one negative eigenvalue, affine multiplicative problems, and fractional multiplicative optimization.This research was accomplished while the second author was a Fellow of the Alexander von Humboldt Foundation at the University of Trier, Trier, Germany.     

An Extension of the Conjugate Directions Method With Orthogonalization to Large-Scale Problems With Bound Constraints

In our previous works a new method of conjugate directions for large-scale unconstrained minimization problems has been presented [1, 2]. In the paper this algorithm is extended to minimization problems with bound constraints. Because the linear minimization along the newly found conjugate vector is not needed for constructing the next conjugate vector and one arbitrarily step-size (not necessarily the optimal one) is calculated along this conjugate direction, we are able to incorporate naturally the bound constraints into the algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)