Robust linear optimization under general norms

We explicitly characterize the robust counterpart of a linear programming problem with uncertainty set described by an arbitrary norm. Our approach encompasses several approaches from the literature and provides guarantees for constraint violation under probabilistic models that allow arbitrary dependencies in the distribution of the uncertain coefficients.     

Integration and propagation of a multi-criteria decision making model in constraint programming

In this paper we propose a general integration scheme for a Multi-Criteria Decision Making model of the Multi-Attribute Utility Theory in Constraint Programming. We introduce the Choquet integral as a general aggregation function for multi-criteria optimization problems and define the Choquet global constraint that propagates this function during the Branch-and-Bound search. Finally the benefits of the propagation of the Choquet constraint are evaluated on the examination timetabling problem.     

Two differential equation systems for equality-constrained optimization

This paper presents two differential systems, involving first and second order derivatives of problem functions, respectively, for solving equality-constrained optimization problems. Local minimizers to the optimization problems are proved to be asymptotically stable equilibrium points of the two differential systems. First, the Euler discrete schemes with constant stepsizes for the two differential systems are presented and their convergence theorems are demonstrated. Second, we construct algorithms in which directions are computed by these two systems and the stepsizes are generated by Armijo line search to solve the original equality-constrained optimization problem. The constructed algorithms and the Runge–Kutta method are employed to solve the Euler discrete schemes and the differential equation systems, respectively. We prove that the discrete scheme based on the differential equation system with the second order information has the locally quadratic convergence rate under the local Lipschitz condition. The numerical results given here show that Runge–Kutta method has better stability and higher precision and the numerical method based on the differential equation system with the second information is faster than the other one.     

Differential Conditions for Constrained Nonlinear Programming via Pareto Optimization

We deal with the differential conditions for local optimality. The conditions that we derive for inequality constrained problems do not require constraint qualifications and are the broadest conditions based on only first-order and second-order derivatives. A similar result is proved for equality constrained problems, although the necessary conditions require the regularity of the equality constraints.     

Geometry of optimality conditions and constraint qualifications: The convex case

The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem. In addition, the badly behaved set of constraints, i.e. the set of constraints which causes problems in the Kuhn—Tucker theory, is isolated and a computational procedure for checking whether a feasible point is regular or not is presented.This research was supported by the National Research Council of Canada and le Gouvernement du Quebec and is part of the author's Ph.D. Dissertation done at McGill University, Montreal, Que., under the guidance of Professor S. Zlobec.     

Generalizations of Slater's constraint qualification for infinite convex programs

In this paper we study constraint qualifications and duality results for infinite convex programs (P) = inf{f(x): g(x) – S, x C}, whereg = (g ₁,g ₂) andS = S ₁ ×S ₂,S _i are convex cones,i = 1, 2,C is a convex subset of a vector spaceX, andf andg _i are, respectively, convex andS _i -convex,i = 1, 2. In particular, we consider the special case whenS ₂ is in afinite dimensional space,g ₂ is affine andS ₂ is polyhedral. We show that a recently introduced simple constraint qualification, and the so-called quasi relative interior constraint qualification both extend to (P), from the special case thatg = g ₂ is affine andS = S ₂ is polyhedral in a finite dimensional space (the so-called partially finite program). This provides generalized Slater type conditions for (P) which are much weaker than the standard Slater condition. We exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong ₂ and the finite dimensionality assumption onS ₂, by a local compactness assumption. We then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption. Our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones.     

Constraint-Based Representations of Statistical Graphs

Abstract

Pictor is an environment for statistical graphics that promotes simple commands for common uses and offers the ability to experiment with whole new paradigms. Pictor describes graphs as graphical objects whose component pieces are related by several sorts of constraints. This article describes in detail the constraint system that Pictor uses.     

Faddeev-Jackiw and the improved methods in quantization of the superconductive system

In this paper, we give the overview on Faddeev-Jackiw method and its improved one, as well as the relative studies recently and realize quantization of the superconductive system by the two methods, we get the same results by the two disposal methods. Furthermore, at convenience of the familiar study in this system, we take it as the application example and compare the two methods by dealing with this system from different aspects, demonstrate the improved Faddeev-Jackiw method is effective and significative, and represent the superiorities of the improved Faddeev-Jackiw method. We show that the improved method may simplify investigations of different complicated constrained systems.     

带模糊约束的最小费用流问题

本文首次提出了带模糊约束的最小费用流问题,建立了相应的数学模型并给出了求解这一模型的有关算法。最后,给出了一个具体实例。     

Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications

We consider a class of optimization problems that is called a mathematical program with vanishing constraints (MPVC for short). This class has some similarities to mathematical programs with equilibrium constraints (MPECs for short), and typically violates standard constraint qualifications, hence the well-known Karush-Kuhn-Tucker conditions do not provide necessary optimality criteria. In order to obtain reasonable first order conditions under very weak assumptions, we introduce several MPVC-tailored constraint qualifications, discuss their relation, and prove an optimality condition which may be viewed as the counterpart of what is called M-stationarity in the MPEC-field.