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1.
This paper presents a canonical duality theory for solving quadratic minimization problems subjected to either box or integer
constraints. Results show that under Gao and Strang’s general global optimality condition, these well-known nonconvex and
discrete problems can be converted into smooth concave maximization dual problems over closed convex feasible spaces without
duality gap, and can be solved by well-developed optimization methods. Both existence and uniqueness of these canonical dual
solutions are presented. Based on a second-order canonical dual perturbation, the discrete integer programming problem is
equivalent to a continuous unconstrained Lipschitzian optimization problem, which can be solved by certain deterministic technique.
Particularly, an analytical solution is obtained under certain condition. A fourth-order canonical dual perturbation algorithm
is presented and applications are illustrated. Finally, implication of the canonical duality theory for the popular semi-definite
programming method is revealed. 相似文献
2.
《Optimization》2012,61(3):235-243
In this paper, we derive an unconstrained convex programming approach to solving convex quadratic programming problems in standard form. Related duality theory is established by using two simple inequalities. An ?-optimal solution is obtained by solving an unconstrained dual convex program. A dual-to-primal conversion formula is also provided. Some preliminary computational results of using a curved search method is included 相似文献
3.
Dag Wedelin 《Annals of Operations Research》1995,57(1):283-301
We present an approximation algorithm for solving large 0–1 integer programming problems whereA is 0–1 and whereb is integer. The method can be viewed as a dual coordinate search for solving the LP-relaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working together with this method. The approximation scheme works by adjusting the costs as little as possible so that the new problem has an integer solution. The degree of approximation is determined by a parameter, and for different levels of approximation the resulting algorithm can be interpreted in terms of linear programming, dynamic programming, and as a greedy algorithm. The algorithm is used in the CARMEN system for airline crew scheduling used by several major airlines, and we show that the algorithm performs well for large set covering problems, in comparison to the CPLEX system, in terms of both time and quality. We also present results on some well known difficult set covering problems that have appeared in the literature. 相似文献
4.
In this paper, we examine duality for fractional programming problems in the face of data uncertainty within the framework
of robust optimization. We establish strong duality between the robust counterpart of an uncertain convex–concave fractional
program and the optimistic counterpart of its conventional Wolfe dual program with uncertain parameters. For linear fractional
programming problems with constraint-wise interval uncertainty, we show that the dual of the robust counterpart is the optimistic
counterpart in the sense that they are equivalent. Our results show that a worst-case solution of an uncertain fractional
program (i.e., a solution of its robust counterpart) can be obtained by solving a single deterministic dual program. In the
case of a linear fractional programming problem with interval uncertainty, such solutions can be found by solving a simple
linear program. 相似文献
5.
An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage
of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes
the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against
that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course
of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development
of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal
feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods
of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example.
This research was supported in part by the Army Research Office under Grant No. DAAG 29-77-G-0114 and in part by the National
Science Foundation under Grant No. MCS-6006065. 相似文献
6.
F. Borrelli A. Bemporad M. Morari 《Journal of Optimization Theory and Applications》2003,118(3):515-540
We propose a novel algorithm for solving multiparametric linear programming problems. Rather than visiting different bases of the associated LP tableau, we follow a geometric approach based on the direct exploration of the parameter space. The resulting algorithm has computational advantages, namely the simplicity of its implementation in a recursive form and an efficient handling of primal and dual degeneracy. Illustrative examples describe the approach throughout the paper. The algorithm is used to solve finite-time constrained optimal control problems for discrete-time linear dynamical systems. 相似文献
7.
Consider a linear programming problem in Karmarkar's standard form. By perturbing its linear objective function with an entropic barrier function and applying generalized geometric programming theory to it, Fang recently proposed an unconstrained convex programming approach to finding an epsilon-optimal solution. In this paper, we show that Fang's derivation of an unconstrained convex dual program can be greatly simplified by using only one simple geometric inequality. In addition, a system of nonlinear equations, which leads to a pair of primal and dual epsilon-optimal solutions, is proposed for further investigation.This work was partially supported by the North Carolina Supercomputing Center and a 1990 Cray Research Grant. The authors are indebted to Professors E. L. Peterson and R. Saigal for stimulating discussions. 相似文献
8.
A dual algorithm based on the smooth function proposed by Polyak (1988) is constructed for solving nonlinear programming problems with inequality constraints. It generates a sequence of points converging locally to a Kuhn-Tucker point by solving an unconstrained minimizer of a smooth potential function with a parameter. We study the relationship between eigenvalues of the Hessian of this smooth potential function and the parameter, which is useful for analyzing the effectiveness of the dual algorithm. 相似文献
9.
Sparse covariance selection problems can be formulated as log-determinant (log-det) semidefinite programming (SDP) problems with large numbers of linear constraints. Standard primal–dual interior-point methods that are based on solving the Schur complement equation would encounter severe computational bottlenecks if they are applied to solve these SDPs. In this paper, we consider a customized inexact primal–dual path-following interior-point algorithm for solving large scale log-det SDP problems arising from sparse covariance selection problems. Our inexact algorithm solves the large and ill-conditioned linear system of equations in each iteration by a preconditioned iterative solver. By exploiting the structures in sparse covariance selection problems, we are able to design highly effective preconditioners to efficiently solve the large and ill-conditioned linear systems. Numerical experiments on both synthetic and real covariance selection problems show that our algorithm is highly efficient and outperforms other existing algorithms. 相似文献
10.
A dual l
p-norm perturbation approach is introduced for solving convex quadratic programming problems. The feasible region of the Lagrangian dual program is approximated by a proper subset that is defined by a single smooth convex constraint involving the l
p-norm of a vector measure of constraint violation. It is shown that the perturbed dual program becomes the dual program as p and, under some standard conditions, the optimal solution of the perturbed dual program converges to a dual optimal solution. A closed-form formula that converts an optimal solution of the perturbed dual program into a feasible solution of the primal convex quadratic program is also provided. Such primal feasible solutions converge to an optimal primal solution as p. The proposed approach generalizes the previously proposed primal perturbation approach with an entropic barrier function. Its theory specializes easily for linear programming. 相似文献
11.
In various applications the search for certificates for certain properties (e.g., stability of dynamical systems, program termination) can be formulated as a quantified constraint solving problem with quantifier prefix exists-forall. In this paper, we present an algorithm for solving a certain class of such problems based on interval techniques in combination with conservative linear programming approximation. In comparison with previous work, the method is more general—allowing general Boolean structure in the input constraint, and more efficient—using splitting heuristics that learn from the success of previous linear programming approximations. 相似文献
12.
Parallel algorithms for nonlinear programming problems 总被引:1,自引:0,他引:1
M. Dayde 《Journal of Optimization Theory and Applications》1989,61(1):23-46
This paper describes several parallel algorithms for solving nonlinear programming problems. Two approaches where parallelism can successfully be introduced have been explored: a quadratic approximation method based on penalty function and a dual method. These methods are improved by using two algorithms originally proposed for solving unconstrained problems: the parallel variable metric algorithm and the parallel Jacobson-Oksman algorithm. Even though general problems are dealt with, particular emphasis is placed on the potential of these parallel methods for separable programming problems. The numerical effectiveness of the algorithms is demonstrated on a set of test problems using a Cray-1S vector computer and serial computers (with respect to sequential versions of the same methods).These studies were sponsored in part by the CERT. The author would particularly like to thank Ph. Berger (LSI-ENSEEIHT), the researchers of the DERI (CERT) and of the Groupe Structures, Aerospatiale, for their assistance. 相似文献
13.
Da Gang Tian 《Journal of Global Optimization》2014,58(1):109-135
We propose an entire space polynomial-time algorithm for linear programming. First, we give a class of penalty functions on entire space for linear programming by which the dual of a linear program of standard form can be converted into an unconstrained optimization problem. The relevant properties on the unconstrained optimization problem such as the duality, the boundedness of the solution and the path-following lemma, etc, are proved. Second, a self-concordant function on entire space which can be used as penalty for linear programming is constructed. For this specific function, more results are obtained. In particular, we show that, by taking a parameter large enough, the optimal solution for the unconstrained optimization problem is located in the increasing interval of the self-concordant function, which ensures the feasibility of solutions. Then by means of the self-concordant penalty function on entire space, a path-following algorithm on entire space for linear programming is presented. The number of Newton steps of the algorithm is no more than $O(nL\log (nL/ {\varepsilon }))$ , and moreover, in short step, it is no more than $O(\sqrt{n}\log (nL/{\varepsilon }))$ . 相似文献
14.
15.
《European Journal of Operational Research》2002,139(3):511-520
We designed an algorithm for the multiparametric 0–1-integer linear programming (ILP) problem with the perturbation of the constraint matrix, the objective function and the right-hand side vector simultaneously considered. Our algorithm works by choosing an appropriate finite sequence of non-parametric mixed integer linear programming (MILP) problems in order to obtain a complete multiparametrical analysis. The algorithm may be implemented by using any software capable of solving MILP problems. 相似文献
16.
We propose an algorithm to solve generalized fractional programming problems. The proposed algorithm combines the parametric approach and the Huard method of centers. A minimum of the problem is obtained by solving a sequence of unconstrained optimization problems. 相似文献
17.
双层线性规划的一个全局优化方法 总被引:7,自引:0,他引:7
用线性规划对偶理论分析了双层线性规划的最优解与下层问题的对偶问题可行域上极点之间的关系,通过求得下层问题的对偶问题可行域上的极点,将双层线性规划转化为有限个线性规划问题,从而用线性规划方法求得问题的全局最优解.由于下层对偶问题可行域上只有有限个极点,所以方法具有全局收敛性. 相似文献
18.
19.
In this paper, we propose unconstrained and constrained posynomial Geometric Programming (GP) problem with negative or positive integral degree of difficulty. Conventional GP approach has been modified to solve some special typer of GP problems. In specific case, when the degree of difficulty is negative, the normality and the orthogonality conditions of the dual program give a system of linear equations. No general solution vector exists for this system of linear equations. But an approximate solution can be determined by the least square and also max-min method. Here, modified form of geometric programming method has been demonstrated and for that purpose necessary theorems have been derived. Finally, these are illustrated by numerical examples and applications. 相似文献
20.
《European Journal of Operational Research》2002,139(1):26-41
Various computational difficulties arise in using decision set-based vector maximization methods to solve multiple objective linear programming problems. As a result, several researchers have begun to explore the possibility of solving these problems by examining subsets of their outcome sets, rather than of their decision sets. In this article, we present and validate a basic weight set decomposition approach for generating the set of all efficient extreme points in the outcome set of a multiple objective linear program. Based upon this approach, we then develop an algorithm, called the Weight Set Decomposition Algorithm, for generating this set. A sample problem is solved using this algorithm, and the main potential computational and practical advantages of the algorithm are indicated. 相似文献