On the finite convergence of interior-point algorithms for linear programming

It has been shown [8] that numerous interior-point algorithms for linear programming (LP) generate solution sequences that converge to strict complementarity solutions, or interior solutions on the optimal face. In this note we further establish a theoretical base for Gay's test (Gay, 1989) to identify the optimal face, and develop a new termination procedure to obtain an exact solution on the optimal face. We also report some numerical results for solving a set of LP test problems, each of which has a highly degenerate and unbounded optimal face.Research supported in part by NSF Grant DDM-8922636, The Iowa Business School Summer Grant, and the Interdisciplinary Research Grant of the University of Iowa Center for Advanced Studies.     

Equivalence among various derivatives and subdifferentials of the distance function

For a nonempty closed set C in a normed linear space X with uniformly Gâteaux differentiable norm, it is shown that the distance function d_C is strictly differentiable at x∈X?C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then d_C is Fréchet differentiable at x∈X?C iff its Fréchet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given.     

Optimality and mixed saddle point criteria in multiobjective optimization

In this paper, higher order strong convexity for Lipschitz functions is introduced and is utilized to derive the optimality conditions for the new concept of strict minimizer of higher order for a multiobjective optimization problem. Variational inequality problem is introduced and its solutions are related to the strict minimizers of higher order for a multiobjective optimization problem. The notion of vector valued partial Lagrangian is also introduced and equivalence of the mixed saddle points of higher order and higher order minima are provided.     

Monotonicity characteristic of Köthe-Bochner spaces

Estimates of the characteristic of monotonicity in Köthe-Bochner function spaces E(X) with some consequences are given. Characterizations of strict and uniform monotonicity of the sequence lattice are obtained.     

Optimality conditions for an isolated minimum of order two in C constrained optimization

In this paper we obtain first and second-order optimality conditions for an isolated minimum of order two for the problem with inequality constraints and a set constraint. First-order sufficient conditions are derived in terms of generalized convex functions. In the necessary conditions we suppose that the data are continuously differentiable. A notion of strongly KT invex inequality constrained problem is introduced. It is shown that each Kuhn-Tucker point is an isolated global minimizer of order two if and only if the problem is strongly KT invex. The article could be considered as a continuation of [I. Ginchev, V.I. Ivanov, Second-order optimality conditions for problems with C¹ data, J. Math. Anal. Appl. 340 (2008) 646-657].     

Cohomology of classical algebraic groups from the functorial viewpoint

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra structures on the (stable) cohomology of a classical algebraic group with coefficients in a Hopf algebra. Our result also opens the way to new explicit cohomology computations. We give an example inspired by recent computations of Djament and Vespa.     

On certain unimodal sequences and strict partitions

Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.     

大规模严格凸二次规划问题一个新算法

根据广义乘子法的思想,将具有等式约束和非负约束的凸二次规划问题转化只有非负约束的简单凸二次规划,通过简单凸二次规划来得到解等式约束一非负约束的凸二次规划新算法,新算法不用求逆矩阵,这样可充分保持矩阵的稀疏性,用来解大规模稀疏问题,数值结果表明：在微机４８６／３３上就能解较大规模的凸二次规划。     

Examples of non-archimedean Fréchet spaces without nuclear Köthe quotients

Let K be a spherically complete non-archimedean valued field. We prove that the dual space l_∞ of the Banach space c₀ has a total strongly non-norming subspace M. Using this subspace M we construct a non-normable Fréchet space F of countable type with a continuous norm such that its strong dual is a strict LB-space. Next we show that F has no nuclear Köthe quotient.     

A Nonmonotone Trust Region Method for Nonlinear Programming with Simple Bound Constraints

In this paper we propose a nonmonotone trust region algorithm for optimization with simple bound constraints. Under mild conditions, we prove the global convergence of the algorithm. For the monotone case it is also proved that the correct active set can be identified in a finite number of iterations if the strict complementarity slackness condition holds, and so the proposed algorithm reduces finally to an unconstrained minimization method in a finite number of iterations, allowing a fast asymptotic rate of convergence. Numerical experiments show that the method is efficient. Accepted 5 September 2000. Online publication 4 December 2000.