First and second order conditions for a class of nondifferentiable optimization problems

This paper gives first and in particular second order necessary and sufficient conditions for a class of nondifferentiable optimization problems in which there are both objective and constraint functions defined in terms of a norm. The conditions are expressed in terms of a Lagrangian function and its derivatives, and use the ideas of feasible directional sequence and subgradients. Certain regularity assumptions are required and for the second order necessary conditions it is shown that the assumption is realistic for polyhedral norms. Illustrative examples are discussed.     

Finite-Dimensional Orthogonality Structures for Hall-Littlewood Polynomials

We present a finite-dimensional system of discrete orthogonality relations for the Hall-Littlewood polynomials. A compact determinantal formula for the weights of the discrete orthogonality measure is formulated in terms of a Gaudin-type conjecture for the normalization constants of a dual system of orthogonality relations. The correctness of our normalization conjecture has been checked in some special cases: for Hall-Littlewood polynomials up to four variables (i), for the reduction to Schur polynomials (ii), and in a continuum limit in which the Hall-Littlewood polynomials degenerate into the Bethe Ansatz eigenfunctions of the Schrödinger operator for identical Bose particles on the circle with pairwise delta-potential interactions (iii).     

The Arboreal Structure of the Metric Space X ≔ GL 2(ℚ p )/ℚ p ∗GL 2(ℤ p )

In this article we define an arboreal structure on the metric space X: = GL ₂(? _p )/? _p ?GL ₂(? _p ) and give an elementary proof of the fact that X is a (p + 1)-valent tree.     

Addendum to “Contractive projections on Banach algebras” [J. Funct. Anal. 254 (10) (2008) 2513-2533]

The purpose of this addendum is to clarify two shortcomings in the paper of the title.     

用最小范数法解三峡梯级水电站短期经济调度问题

本文主要研究三峡梯级水电站与华中、华东和川东电网联网的短期经济调度问题,利用泛函分析和运筹学相结合的方法建立了三峡梯级水电站日负荷最优分配的数学模型。本文扩充和推广了Hawary和Christensen的最小范数法用来求解这个具有等式和不等式约束的高维非线性含时滞的动态最优化问题,最优策略由一组动态的非线性代数、微分方程确定。引入适当的变量并进行适当化简,最终可将三峡梯级水电系统的经济调度问题转化为一个最小范数问题,并给出了最优解的具体表达式.用Lagrange乘子和Kuhn-Tucker乘子将约束条件并入目标函数中形成一个增广价格函数。通过变换可将该无约束优化问题转化为求解非线性代数方程组的问题。本文选用Fletcher-Reeves共轭梯度法求解无约束极值问题.在IBM-PC型微机上进行了试算。试算结果表明用最小范数法求解三峡梯级水电站日负荷最优分配问题是完全可行的,梯级水耗率有明显下降,能获得一定的经济效益。     

A generalization of the Frank—Wolfe theorem

The Frank—Wolfe theorem states that a quadratic function, bounded below on a nonempty polyhedral convex set, attains its infimum there. This paper gives sufficient conditions under which a function either attains its infimum on a nonempty polyhedral convex set or is unbounded below on some halfline of that set. Quadratic functions are shown to satisfy these sufficient conditions.Research and reproduction of this report were partially supported by the National Science Foundation Grant MCS76-81259; and the Office of Naval Research Contract N00014-75-C-0267.     

Pinchings and Norms of Scaled Triangular Matrices

Suppose U is an upper-triangular matrix, and D a nonsingular diagonal matrix whose diagonal entries appear in nondescending order of magnitude down the diagonal. It is proved that $$|D^{-1}UD|ge|U|$$ for any matrix norm that is reduced by a pinching. In addition to known examples -weakly unitarily invariant norms - we show that any matrix norm defined by $$| A |^{underline{underline {{rm def}}} } mathop {max }limits_{x ne 0,y ne 0} {{{mathop{rm Re}nolimits} (x^*Ay)} over {phi (x)psi (y)}},$$ where θ (.) and y (.) are two absolute vector norms, has this property. This includes l p operator norms as a special case.     

Rank Reducing Matrix Norms

We consider approximation numbers for some norms on matrices, and look at the question when a closest rank h p approximant can be chosen to reduce the rank of a matrix by p . If the latter is always possible, we call the norm rank p reducing. It is easily seen that any unitarily invariant norm is rank p reducing. We show that any absolute norm on $\shadC^{n \times m}$ is rank n m 1 reducing and that the numerical radius norm on $ \shadC^{n\times n}$ is rank n m 1 reducing as well. Non-examples and computations of approximation numbers are also presented.