解带有二次约束二次规划的一个整体优化方法

在本文中，我们提出了一种解带有二次约束二次规划问题（QP）的新算法，这种方法是基于单纯形分枝定界技术，其中包括极小极大问题和线性规划问题作为子问题，利用拉格朗日松弛和投影次梯度方法来确定问题（QP）最优值的下界，在问题（QP）的可行域是n维的条件下，如果这个算法有限步后终止，得到的点必是问题（QP）的整体最优解；否则，该算法产生的点的序列{v^k}的每一个聚点也必是问题（QP）的整体最优解。     

The Euclidean Steiner tree problem in Rn: A mathematical programming formulation

A nonconvex mixed-integer programming formulation for the Euclidean Steiner Tree Problem (ESTP) in Rⁿ is presented. After obtaining separability between integer and continuous variables in the objective function, a Lagrange dual program is proposed. To solve this dual problem (and obtaining a lower bound for ESTP) we use subgradient techniques. In order to evaluate a subgradient at each iteration we have to solve three optimization problems, two in polynomial time, and one is a special convex nondifferentiable programming problem. This revised version was published online in June 2006 with corrections to the Cover Date.     

Evaluating mixed-integer programming models over multiple right-hand sides

A critical measure of model quality for a mixed-integer program (MIP) is the difference, or gap, between its optimal objective value and that of its linear programming relaxation. In some cases, the right-hand side is not known exactly; however, there is no consensus metric for evaluating a MIP model when considering multiple right-hand sides. In this paper, we provide model formulations for the expectation and extrema of absolute and relative MIP gap functions over finite discrete sets.     

Multiobjective Control Approximation Problems: Duality and Optimality

A general convex multiobjective control approximation problem is considered with respect to duality. The single objectives contain linear functionals and powers of norms as parts, measuring the distance between linear mappings of the control variable and the state variables. Moreover, linear inequality constraints are included. A dual problem is established, and weak and strong duality properties as well as necessary and sufficient optimality conditions are derived. Point-objective location problems and linear vector optimization problems turn out to be special cases of the problem investigated. Therefore, well-known duality results for linear vector optimization are obtained as special cases.     

Absolute value programming

We investigate equations, inequalities and mathematical programs involving absolute values of variables such as the equation Ax+B|x| = b, where A and B are arbitrary m× n real matrices. We show that this absolute value equation is NP-hard to solve, and that solving it with B = I solves the general linear complementarity problem. We give sufficient optimality conditions and duality results for absolute value programs as well as theorems of the alternative for absolute value inequalities. We also propose concave minimization formulations for absolute value equations that are solved by a finite succession of linear programs. These algorithms terminate at a local minimum that solves the absolute value equation in almost all solvable random problems tried.     

On the acceleration of the double smoothing technique for unconstrained convex optimization problems

In this article, we investigate the possibilities of accelerating the double smoothing (DS) technique when solving unconstrained nondifferentiable convex optimization problems. This approach relies on the regularization in two steps of the Fenchel dual problem associated with the problem to be solved into an optimization problem having a differentiable strongly convex objective function with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method. The aim of this article is to show how the properties of the functions in the objective of the primal problem influence the implementation of the DS approach and its rate of convergence. The theoretical results are applied to linear inverse problems by making use of different regularization functionals.     

A canonical dual approach for solving linearly constrained quadratic programs

This paper provides a canonical dual approach for minimizing a general quadratic function over a set of linear constraints. We first perturb the feasible domain by a quadratic constraint, and then solve a “restricted” canonical dual program of the perturbed problem at each iteration to generate a sequence of feasible solutions of the original problem. The generated sequence is proven to be convergent to a Karush-Kuhn-Tucker point with a strictly decreasing objective value. Some numerical results are provided to illustrate the proposed approach.     

Global minimization by reducing the duality gap

We derive a general principle demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated. The principle can be implemented in a Branch and Bound algorithm which computes an approximate global solution and a corresponding lower bound on the global optimal value. The algorithm involves decomposition and a nonsmooth local search. Numerical results for applying the algorithm to the pooling problem in oil refineries are given.Research supported by Shell Laboratorium, Amsterdam, and GIF—The German—Israel Foundation for Scientific Research and Development.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

A new reformulation-linearization technique for bilinear programming problems

This paper is concerned with the development of an algorithm for general bilinear programming problems. Such problems find numerous applications in economics and game theory, location theory, nonlinear multi-commodity network flows, dynamic assignment and production, and various risk management problems. The proposed approach develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm. The method first reformulates the problem by constructing a set of nonnegative variable factors using the problem constraints, and suitably multiplies combinations of these factors with the original problem constraints to generate additional valid nonlinear constraints. The resulting nonlinear program is subsequently linearized by defining a new set of variables, one for each nonlinear term. This RLT process yields a linear programming problem whose optimal value provides a tight lower bound on the optimal value to the bilinear programming problem. Various implementation schemes and constraint generation procedures are investigated for the purpose of further tightening the resulting linearization. The lower bound thus produced theoretically dominates, and practically is far tighter, than that obtained by using convex envelopes over hyper-rectangles. In fact, for some special cases, this process is shown to yield an exact linear programming representation. For the associated branch-and-bound algorithm, various admissible branching schemes are discussed, including one in which branching is performed by partitioning the intervals for only one set of variables x or y, whichever are fewer in number. Computational experience is provided to demonstrate the viability of the algorithm. For a large number of test problems from the literature, the initial bounding linear program itself solves the underlying bilinear programming problem.This paper was presented at the II. IIASA Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.     

A proximal point algorithm for generalized fractional programs

In this paper, the problem of solving generalized fractional programs will be addressed. This problem has been extensively studied and several algorithms have been proposed. In this work, we propose an algorithm that combines the proximal point method with a continuous min–max formulation of discrete generalized fractional programs. The proposed method can handle non-differentiable convex problems with possibly unbounded feasible constraints set, and solves at each iteration a convex program with unique dual solution. It generates two sequences that approximate the optimal value of the considered problem from below and from above at each step. For a class of functions, including the linear case, the convergence rate is at least linear.     

下层以最优值反应上层的两层线性规则

本文证明了下层以最优值反应上层的两层线性规划可转化为一线性Max-min问题，进而得出其与一双线性规划问题等价。基于此结论可以讨论这种特殊两层问题的几何性质，最优性条件及算法设计。     

Nonlinear Lagrangian Theory for Nonconvex Optimization

The Lagrangian function in the conventional theory for solving constrained optimization problems is a linear combination of the cost and constraint functions. Typically, the optimality conditions based on linear Lagrangian theory are either necessary or sufficient, but not both unless the underlying cost and constraint functions are also convex.We propose a somewhat different approach for solving a nonconvex inequality constrained optimization problem based on a nonlinear Lagrangian function. This leads to optimality conditions which are both sufficient and necessary, without any convexity assumption. Subsequently, under appropriate assumptions, the optimality conditions derived from the new nonlinear Lagrangian approach are used to obtain an equivalent root-finding problem. By appropriately defining a dual optimization problem and an alternative dual problem, we show that zero duality gap will hold always regardless of convexity, contrary to the case of linear Lagrangian duality.     

A dynamic programming method for controlled linear systems with delays

A linear system with permanent delay is considered. A method of dynamic programming for constructing attainability sets and solving the problem of target control for the systems is used. The expressions for value functionals described by solutions to the corresponding Hamilton-Jacobi-Bellman equation are obtained. It is proved that these value functionals calculated by means of convex analysis satisfy the above equations. Strategies for synthesized control for the problem of hitting on the target set are given.     

Controlled perturbations for quadratically constrained quadratic programs

Consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function. The duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear. However, the objective function is not differentiable over the constraint region. In this paper, we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem. Based on this relationship, we establish a perturbation theory to display that a well-controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an ε-optimal dual solution for an arbitrarily small number ε. A simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding ε-optimal primal solution. Moreover, a numerical example is included to illustrate the potential of this controlled perturbation scheme.     

Uniform LP duality for semidefinite and semi-infinite programming

Recently, a semidefinite and semi-infinite linear programming problem (SDSIP), its dual (DSDSIP), and uniform LP duality between (SDSIP) and (DSDSIP) were proposed and studied by Li et al. (Optimization 52:507–528, 2003). In this paper, we show that (SDSIP) is an ordinary linear semi-infinite program and, therefore, all the existing results regarding duality and uniform LP duality for linear semi-infinite programs can be applied to (SDSIP). By this approach, the main results of Li et al. (Optimization 52:507–528, 2003) can be obtained easily.     

Subdifferentiability and the Duality Gap

We point out a connection between sensitivity analysis and the fundamental theorem of linear programming by characterizing when a linear programming problem has no duality gap. The main result is that the value function is subdifferentiable at the primal constraint if and only if there exists an optimal dual solution and there is no duality gap. To illustrate the subtlety of the condition, we extend Kretschmer's gap example to construct (as the value function of a linear programming problem) a convex function which is subdifferentiable at a point but is not continuous there. We also apply the theorem to the continuum version of the assignment model.     

A necessary and sufficient condition for duality in control problems: KT-invex functionals

In this article, we introduce a new condition on functionals of a control problem, and for that purpose we define the KT-invex functionals. We extend recent optimality control works to the study of duality. In this way we establish weak, strong and converse duality results under KT-invexity. Furthermore, we prove that KT-invexity is not only a sufficient condition for establishing duality, but it is necessary.     

Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods

A family of complementarity problems is defined as extensions of the well-known linear complementarity problem (LCP). These are:

A finite,nonadjacent extreme-point search algorithm for optimization over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple-objective linear program serves many useful purposes in multiple-criteria decision making. Mathematically, problem (P) can be classified as a global optimization problem. Such problems are much more difficult to solve than convex programming problems. In this paper, a nonadjacent extreme-point search algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact extreme-point optimal solution for the problem after a finite number of iterations. It can be implemented using only linear programming methods. Convergence of the algorithm is proven, and a discussion is included of its main advantages and disadvantages.The author owes thanks to two anonymous referees for their helpful comments.