Generation of Degenerate Linear Programming Problems

We present a method for constructing linear programming problems with randomly generated data. Besides the number of variables and constraints, the dimensions of the primal and dual faces are given. We show that, for problems in which the constraint matrix is carelessly constructed with random entries, with probability one only one between primal degeneracy and dual degeneracy appears.     

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．     

Post-optimality sensitivity analysis in abstract spaces with applications to continuous-time programming problems

The programming problem under consideration consists in maximizing a concave objective functional, subject to convex operator inequality contraints. The assumptions include the existence of an optimum solution, Fréchet differentiability of all operators involved, and the existence of the topological complement of the null space of the Fréchet derivative of the constraint operator. It is shown that the rate of change of the optimum value of the objective functional due to the perturbation is measured by the dual. The optimum values of the primal variables are locally approximated as linear functions of the perturbation; the theory of generalized inverse operators is used in the approximation. We give an approximation to the primal variables if the problem is perturbed. The results are specialized for some continuous-time and finite-dimensional cases. Two examples for finite-dimensional problems are given. We apply the theory to the continuous-time linear programming problem and prove some continuity results for the optimal primal and dual objective functionals.The authors are indebted to the Natural Sciences and Engineering Research Council of Canada for financial support through Grants A4109 and A7329, respectively. They would also like to thank the referee for his comments.     

Computational Experience with Ill-Posed Problems in Semidefinite Programming

Many theoretical and algorithmic results in semidefinite programming are based on the assumption that Slater's constraint qualification is satisfied for the primal and the associated dual problem. We consider semidefinite problems with zero duality gap for which Slater's condition fails for at least one of the primal and dual problem. We propose a numerically reasonable way of dealing with such semidefinite programs. The new method is based on a standard search direction with damped Newton steps towards primal and dual feasibility.     

Application of the alternating direction method of multipliers to separable convex programming problems

This paper presents a decomposition algorithm for solving convex programming problems with separable structure. The algorithm is obtained through application of the alternating direction method of multipliers to the dual of the convex programming problem to be solved. In particular, the algorithm reduces to the ordinary method of multipliers when the problem is regarded as nonseparable. Under the assumption that both primal and dual problems have at least one solution and the solution set of the primal problem is bounded, global convergence of the algorithm is established.     

On Regularization Methods for Inverse Problems of Dynamic Type

In this paper, we consider new regularization methods for linear inverse problems of dynamic type. These methods are based on dynamic programming techniques for linear quadratic optimal control problems. Two different approaches are followed: a continuous and a discrete one. We prove regularization properties and also obtain rates of convergence for the methods derived from both approaches. A numerical example concerning the dynamic EIT problem is used to illustrate the theoretical results.     

Properties of the Central Points in Linear Programming Problems

In this paper we list several useful properties of central points in linear programming problems. We study the logarithmic barrier function, the analytic center and the central path, relating the proximity measures and scaled Euclidean distances defined for the primal and primal–dual problems. We study the Newton centering steps, and show how large the short steps used in path following algorithms can actually be, and what variation can be ensured for the barrier function in each iteration of such methods. We relate the primal and primal–dual Newton centering steps and propose a primal-only path following algorithm for linear programming.     

A Variable Krasnoselski–Mann Algorithm for a New Class of Fixed Point Problems

We study the convergence of a variable version of the Krasnoselski–Mann algorithm applied to a primal dual fixed point problem. The link with Spingarn's partial inverse method is made, and an application to feasibility problems and mathematical programming is also proposed.     

Feasibility and solvability of Lyapunov-type linear programming over symmetric cones

In this paper, we consider the Lyapunov-type linear programming and its dual over symmetric cones. By introducing and characterizing the generalized inverse of Lyapunov operator in Euclidean Jordan algebras, we establish two kinds of Lyapunov-type Farkas’ lemmas to exhibit feasibilities of the corresponding primal and dual programming problems, respectively. As one of the main results, we show that the feasibilities of the primal and dual problems lead to the solvability of the primal problem and zero duality gap under some mild condition. In this case, we obtain that any solution to the pair of primal and dual problems is equivalent to the solution of the corresponding KKT system.     

Optimality and duality for nonsmooth multiobjective fractional programming with mixed constraints

We consider nonsmooth multiobjective fractional programming problems with inequality and equality constraints. We establish the necessary and sufficient optimality conditions under various generalized invexity assumptions. In addition, we formulate a mixed dual problem corresponding to primal problem, and discuss weak, strong and strict converse duality theorems. This research was partially supported by Project no. 850203 and Center of Excellence for Mathematics, University of Isfahan, Iran.     

Dual gradient method for linearly constrained,strongly convex,separable mathematical programming problems

A new dual gradient method is given to solve linearly constrained, strongly convex, separable mathematical programming problems. The dual problem can be decomposed into one-dimensional problems whose solutions can be computed extremely easily. The dual objective function is shown to have a Lipschitz continuous gradient, and therefore a gradient-type algorithm can be used for solving the dual problem. The primal optimal solution can be obtained from the dual optimal solution in a straightforward way. Convergence proofs and computational results are given.     

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.     

On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian*

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. *Partially Supported by 2003 UniSA ITEE Small Research Grant Ero2. Supported by CAPES, Brazil, Grant No. 0664-02/2, during her visit to the School of Mathematics and Statistics, UniSA.     

Minimization of Linear Functionals Defined on Solutions of Large-Scale Discrete Ill-Posed Problems

The minimization of linear functionals defined on the solutions of discrete ill-posed problems arises, e.g., in the computation of confidence intervals for these solutions. In 1990, Eldén proposed an algorithm for this minimization problem based on a parametric programming reformulation involving the solution of a sequence of trust-region problems, and using matrix factorizations. In this paper, we describe MLFIP, a large-scale version of this algorithm where a limited-memory trust-region solver is used on the subproblems. We illustrate the use of our algorithm in connection with an inverse heat conduction problem. AMS subject classification (2000) 65F22     

ON SOME PROPERTIES OF SOLUTIONS TO SEMIDEFINITE PROGRAMMING

It is well known that for symmetric linear programming there exists a strictly complementary solution if the primal and the dual problems are both feasible. However, this is not necessary true for symmetric or general semide finite programming even if both the primal problem and its dual problem are strictly feasible. Some other properties are also concerned.     

半定规划的限制逆问题与广义逆问题

本文提出了半定规划的限制逆问题与广义逆问题，利用半定规划的最优性条件，分别给出了其在l∞，l1,l2模意义下的数学模型，它们仍为半定规划问题。     

On the Solution of Inverse Problems for Generalized Oxygen Consumption

We present the solution of some inverse problems for one-dimensional free boundary problems of oxygen consumption type, with a semilinear convection-diffusion-reaction parabolic equation. Using a fixed domain transformation (Landau's transformation) the direct problem is reduced to a system of ODEs. To minimize the objective functionals in the inverse problems, we approximate the data by a finite number of parameters with respect to which automatic differentiation is applied.     

Robust duality for generalized convex programming problems under data uncertainty

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. A robust strong duality theorem is given whenever the Lagrangian function is convex. We provide classes of uncertain non-convex programming problems for which robust strong duality holds under a constraint qualification. In particular, we show that robust strong duality is guaranteed for non-convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized Slater condition. Numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems. We further show that robust duality continues to hold under a weakened convexity condition.     

Duality Theory in Interval-Valued Linear Programming Problems

The weak and strong duality theorems in interval-valued linear programming problems are derived in this paper. The primal and dual interval-valued linear programming problems are formulated by proposing the concept of a scalar (inner) product of closed intervals. We introduce a solution concept that is essentially similar to the notion of nondominated solution in multiobjective programming problems by imposing a partial ordering on the set of all closed intervals. Under these settings, the weak and strong duality theorems for interval-valued linear programming problems are derived naturally.     

Duality Theory in Fuzzy Optimization Problems

A solution concept of fuzzy optimization problems, which is essentially similar to the notion of Pareto optimal solution (nondominated solution) in multiobjective programming problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. We also introduce a concept of fuzzy scalar (inner) product based on the positive and negative parts of fuzzy numbers. Then the fuzzy-valued Lagrangian function and the fuzzy-valued Lagrangian dual function for the fuzzy optimization problem are proposed via the concept of fuzzy scalar product. Under these settings, the weak and strong duality theorems for fuzzy optimization problems can be elicited. We show that there is no duality gap between the primal and dual fuzzy optimization problems under suitable assumptions for fuzzy-valued functions.