有界整规划中的渐近强非线性对偶

本文提出了一种整数规划中的指数一对数对偶.证明了此指数-对数对偶方法具有的渐近强对偶性质,并提出了不需要进行对偶搜索来解原整数规划问题的方法.特别地,当选取合适的参数和对偶变量时,原整数规划问题的解可以通过解一个非线性松弛问题来得到.对具有整系数目标函数及约束函数的多项式整规划问题,给出了参数及对偶变量的取法.     

Arbitrary-norm separating plane

A plane separating two point sets in n-dimensional real space is constructed such that it minimizes the sum of arbitrary-norm distances of misclassified points to the plane. In contrast to previous approaches that used surrogates for distance-minimization, the present work is based on a precise norm-dependent explicit closed form for the projection of a point on a plane. This projection is used to formulate the separating-plane problem as a minimization of a convex function on a unit sphere in a norm dual to that of the arbitrary norm used. For the 1-norm, the problem can be solved in polynomial time by solving 2n linear programs or by solving a bilinear program. For a general p-norm, the minimization problem can be transformed via an exact penalty formulation to minimizing the sum ofa convex function and a bilinear function on a convex set. For the one and infinity norms, a finite successive linearization algorithm can be used for solving the exact penalty formulation.     

Value-Estimation Function Method for Constrained Global Optimization

A novel value-estimation function method for global optimization problems with inequality constraints is proposed in this paper. The value-estimation function formulation is an auxiliary unconstrained optimization problem with a univariate parameter that represents an estimated optimal value of the objective function of the original optimization problem. A solution is optimal to the original problem if and only if it is also optimal to the auxiliary unconstrained optimization with the parameter set at the optimal objective value of the original problem, which turns out to be the unique root of a basic value-estimation function. A logarithmic-exponential value-estimation function formulation is further developed to acquire computational tractability and efficiency. The optimal objective value of the original problem as well as the optimal solution are sought iteratively by applying either a generalized Newton method or a bisection method to the logarithmic-exponential value-estimation function formulation. The convergence properties of the solution algorithms guarantee the identification of an approximate optimal solution of the original problem, up to any predetermined degree of accuracy, within a finite number of iterations.     

A Newton Method for Linear Programming

A fast Newton method is proposed for solving linear programs with a very large (10⁶) number of constraints and a moderate (10²) number of variables. Such linear programs occur in data mining and machine learning. The proposed method is based on the apparently overlooked fact that the dual of an asymptotic exterior penalty formulation of a linear program provides an exact least 2-norm solution to the dual of the linear program for finite values of the penalty parameter but not for the primal linear program. Solving the dual problem for a finite value of the penalty parameter yields an exact least 2-norm solution to the dual, but not a primal solution unless the parameter approaches zero. However, the exact least 2-norm solution to the dual problem can be used to generate an accurate primal solution if mn and the primal solution is unique. Utilizing these facts, a fast globally convergent finitely terminating Newton method is proposed. A simple prototype of the method is given in eleven lines of MATLAB code. Encouraging computational results are presented such as the solution of a linear program with two million constraints that could not be solved by CPLEX 6.5 on the same machine.     

Strong formulations for quadratic optimization with M-matrices and indicator variables

We propose a DC (Difference of two Convex functions) formulation approach for sparse optimization problems having a cardinality or rank constraint. With the largest-k norm, an exact DC representation of the cardinality constraint is provided. We then transform the cardinality-constrained problem into a penalty function form and derive exact penalty parameter values for some optimization problems, especially for quadratic minimization problems which often appear in practice. A DC Algorithm (DCA) is presented, where the dual step at each iteration can be efficiently carried out due to the accessible subgradient of the largest-k norm. Furthermore, we can solve each DCA subproblem in linear time via a soft thresholding operation if there are no additional constraints. The framework is extended to the rank-constrained problem as well as the cardinality- and the rank-minimization problems. Numerical experiments demonstrate the efficiency of the proposed DCA in comparison with existing methods which have other penalty terms.     

A dual exact penalty formulation for the linear complementarity problem

We consider a dual exact penalty formulation for the monotone linear complementarity problem. Tihonov regularization is then used to reduce the solution of the problem to the solution of a sequence of positive-definite, symmetric quadratic programs. A modified form of an SOR method due to Mangasarian is proposed to solve these quadratic programs. We also indicate how to obtain approximate solutions to predefined tolerance by solving a single quadratic program, in special cases.This research was sponsored by US Army Contract DAAG29-80-C-0041, by National Science Foundation Grants DCR-8420963 and MCS-8102684, and AFSOR Grant AFSOR-ISSA-85-0880.     

The exact penalty map for nonsmooth and nonconvex optimization

Augmented Lagrangian duality provides zero duality gap and saddle point properties for nonconvex optimization. On the basis of this duality, subgradient-like methods can be applied to the (convex) dual of the original problem. These methods usually recover the optimal value of the problem, but may fail to provide a primal solution. We prove that the recovery of a primal solution by such methods can be characterized in terms of (i) the differentiability properties of the dual function and (ii) the exact penalty properties of the primal-dual pair. We also connect the property of finite termination with exact penalty properties of the dual pair. In order to establish these facts, we associate the primal-dual pair to a penalty map. This map, which we introduce here, is a convex and globally Lipschitz function and its epigraph encapsulates information on both primal and dual solution sets.     

Misclassification minimization

The problem of minimizing the number of misclassified points by a plane, attempting to separate two point sets with intersecting convex hulls inn-dimensional real space, is formulated as a linear program with equilibrium constraints (LPEC). This general LPEC can be converted to an exact penalty problem with a quadratic objective and linear constraints. A Frank-Wolfe-type algorithm is proposed for the penalty problem that terminates at a stationary point or a global solution. Novel aspects of the approach include: (i) A linear complementarity formulation of the step function that counts misclassifications, (ii) Exact penalty formulation without boundedness, nondegeneracy or constraint qualification assumptions, (iii) An exact solution extraction from the sequence of minimizers of the penalty function for a finite value of the penalty parameter for the general LPEC and an explicitly exact solution for the LPEC with uncoupled constraints, and (iv) A parametric quadratic programming formulation of the LPEC associated with the misclassification minimization problem.This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grants CCR-9101801 and CDA-9024618.     

Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.     

An optimization algorithm for a penalized knapsack problem

We study a variation of the knapsack problem in which each item has a profit, a weight and a penalty; the sum of profits of the selected items minus the largest penalty associated with the selected items must be maximized. We present an ILP formulation and an exact optimization algorithm.     

Deriving the Properties of Linear Bilevel Programming via a Penalty Function Approach

For the linear bilevel programming problem, we propose an assumption weaker than existing assumptions, while achieving similar results via a penalty function approach. The results include: equivalence between (i) existence of a solution to the problem, (ii) existence of an exact penalty function approach for solving the problem, and (iii) achievement of the optimal value of the equivalent form of the problem at some vertex of a certain polyhedral convex set. We prove that the assumption is both necessary and sufficient for the linear bilevel programming problem to admit an exact penalty function formulation, provided that the equivalent form of the problem has a feasible solution. A method is given for computing the minimal penalty function parameter value. This method can be executed by solving a set of linear programming problems. Lagrangian duality is also presented.     

Exact penalty functions in nonlinear programming

It is shown that the existence of a strict local minimum satisfying the constraint qualification of [16] or McCormick's [12] second order sufficient optimality condition implies the existence of a class of exact local penalty functions (that is ones with a finite value of the penalty parameter) for a nonlinear programming problem. A lower bound to the penalty parameter is given by a norm of the optimal Lagrange multipliers which is dual to the norm used in the penalty function.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS74-20584 A02.     

一个新的解线性规划的神经网络

本文提出一个新的解线性规划的Hopfields-型网络。该网络基于线性规划的对偶理论,并使用了Sigmoid函数,但不需要预先给定的罚参数和乘法模拟器,我们证明该网络不仅全局收敛到线性规划的精确解,而且能同时解原规划和对偶规划。由于在该网络中没有使用乘法模拟器而利用了Sigmoid函数,因此该模型是很容易用硬件实现的。     

Augmented Lagrangian Objective Penalty Function

Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.     

New Uniform Parametric Error Bounds

A uniform parametric error bound is a uniform error estimate for feasible solutions of a family of parametric mathematical programming problems. It has been proven useful in exact penalty formulation for bilevel programming problems. In this paper, we derive new sufficient conditions for the existence of uniform parametric error bounds.     

Augmented Lagrangian function, non-quadratic growth condition and exact penalization

In this paper, under the assumption that the perturbation function satisfies a growth condition, necessary and sufficient conditions for an exact penalty representation and a zero duality gap property between the primal problem and its augmented Lagrangian dual problem are established.     

Log-Sigmoid Multipliers Method in Constrained Optimization

In this paper we introduced and analyzed the Log-Sigmoid (LS) multipliers method for constrained optimization. The LS method is to the recently developed smoothing technique as augmented Lagrangian to the penalty method or modified barrier to classical barrier methods. At the same time the LS method has some specific properties, which make it substantially different from other nonquadratic augmented Lagrangian techniques.We established convergence of the LS type penalty method under very mild assumptions on the input data and estimated the rate of convergence of the LS multipliers method under the standard second order optimality condition for both exact and nonexact minimization.Some important properties of the dual function and the dual problem, which are based on the LS Lagrangian, were discovered and the primal–dual LS method was introduced.     

An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm.      

Duality and Exact Penalization for General Augmented Lagrangians

We consider a problem of minimizing an extended real-valued function defined in a Hausdorff topological space. We study the dual problem induced by a general augmented Lagrangian function. Under a simple set of assumptions on this general augmented Lagrangian function, we obtain strong duality and existence of exact penalty parameter via an abstract convexity approach. We show that every cluster point of a sub-optimal path related to the dual problem is a primal solution. Our assumptions are more general than those recently considered in the related literature.