A characteristic condition for convergence of steepest descent approximation to accretive operator equations

In the present paper a sufficient and necessary condition for convergence of steepest descent approximation to accretive operator equations is established, and for the sufficiency part a specific error estimation is also given.     

Convergence of the steepest descent method for minimizing quasiconvex functions

To minimize a continuously differentiable quasiconvex functionf: ⁿ , Armijo's steepest descent method generates a sequencex ^k+1 =x ^k –t _k f(x ^k ), wheret _k >0. We establish strong convergence properties of this classic method: either , s.t. ; or arg minf = , x ^k andf(x ^k ) inff. We also discuss extensions to other line searches.The research of the first author was supported by the Polish Academy of Sciences. The second author acknowledges the support of the Department of Industrial Engineering, Hong Kong University of Science and Technology.We wish to thank two anonymous referees for their valuable comments. In particular, one referee has suggested the use of quasiconvexity instead of convexity off.     

The self regulation problem as an inexact steepest descent method for multicriteria optimization

In this paper we study an inexact steepest descent method for multicriteria optimization whose step-size comes with Armijo’s rule. We show that this method is well-defined. Moreover, by assuming the quasi-convexity of the multicriteria function, we prove full convergence of any generated sequence to a Pareto critical point. As an application, we offer a model for the Psychology’s self regulation problem, using a recent variational rationality approach.     

A novel hybrid immune algorithm and its convergence based on the steepest descent algorithm

This paper proposes a novel hybrid immune algorithm (HIA) that can overcome the typical drawback of the artificial immune algorithm (AIA), which runs slowly and experiences slow convergence. The HIA combines the adaptive AIA based on the steepest descent algorithm. The HIA fully displays global search ability and the global convergence of the immune algorithm. At the same time, it inserts a quasi-descent operator to strengthen its local search ability. A good convergence of the HIA with the quasi-descent idea is shown as well. Numerical experiment results show that the HIA successfully improves running speed and convergence performance.     

Local convergence of the steepest descent method in Hilbert spaces

The aim of this paper is to establish the local convergence of the steepest descent method for C¹-functionals defined on an infinite-dimensional Hilbert space H, under a Palais–Smale-type condition. The functionals f under consideration are also assumed to have a locally Lipschitz continuous gradient operator f. Our approach is based on the solutions of the ordinary differential equation .     

半监督度量学习内蕴最速下降算法的收敛性分析

主要研究对称正定矩阵群上的内蕴最速下降算法的收敛性问题.首先针对一个可转化为对称正定矩阵群上无约束优化问题的半监督度量学习模型,提出对称正定矩阵群上一种自适应变步长的内蕴最速下降算法.然后利用李群上的光滑函数在任意一点处带积分余项的泰勒展开式,证明所提算法在对称正定矩阵群上是线性收敛的.最后通过在分类问题中的数值实验说明算法的有效性.     

On steepest descent curves for quasi convex families in Rn

A connected, linearly ordered subset $\gamma ?{\mathbb{R}}^n$ satisfying $$x_1,x_2,x_3\in \gamma ,\text{and}{x}_1?x_2?x_3\to |x_2?x_1|\le |x_3?x_1|$$ is shown to be a rectifiable curve; a priori bounds for its length are given; moreover, these curves are generalized steepest descent curves of suitable quasi convex functions. Properties of quasi convex families are considered; special curves related to quasi convex families are defined and studied; they are generalizations of steepest descent curves for quasi convex functions and satisfy the previous property. Existence, uniqueness, stability results and length bounds are proved for them.     

Moments of an exponential functional of random walks and permutations with given descent sets

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ₁ + ξ₁ξ₂ + ξ₁ξ₂ξ₃ + ⋯ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μ _k = E(ξ ^k ) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle. This revised version was published online in August 2006 with corrections to the Cover Date.     

A combination of potential reduction steps and steepest descent steps for solving convex programming problems

This paper studies the possibility of combining interior point strategy with a steepest descent method when solving convex programming problems, in such a way that the convergence property of the interior point method remains valid but many iterations do not request the solution of a system of equations. Motivated by this general idea, we propose a hybrid algorithm which combines a primal–dual potential reduction algorithm with the use of the steepest descent direction of the potential function. The complexity of the potential reduction algorithm remains valid but the overall computational cost can be reduced. Our numerical experiments are also reported. Copyright © 2002 John Wiley & Sons, Ltd.     

Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow

We obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow.     

Nonosculating Sets of Positive Reach

For integral geometric and local topological properties of two sets of positive reach, one fixed and the other moving or translating, certain exceptional relative positions have to be excluded. We give a measure geometric justification in form of a Sard-type theorem which extends to more general singular sets. A slightly stronger result has been proved by Schneider for the special case of convex bodies.     

First fall degree and Weil descent

Polynomial systems arising from a Weil descent have many applications in cryptography, including the HFE cryptosystem and the elliptic curve discrete logarithm problem over small characteristic fields. Understanding the exact complexity of solving these systems is essential for the applications. A first step in that direction is to study the first fall degree of the systems. In this paper, we establish a rigorous general bound on the first fall degree of polynomial systems arising from a Weil descent. We also provide experimental data to study the tightness of our bound in general and its plausible consequences on the complexity of polynomial systems arising from a Weil descent.     

On Set Covariance and Three-Point Test Sets

The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic is computed.     

Mirror descent and nonlinear projected subgradient methods for convex optimization

The mirror descent algorithm (MDA) was introduced by Nemirovsky and Yudin for solving convex optimization problems. This method exhibits an efficiency estimate that is mildly dependent in the decision variables dimension, and thus suitable for solving very large scale optimization problems. We present a new derivation and analysis of this algorithm. We show that the MDA can be viewed as a nonlinear projected-subgradient type method, derived from using a general distance-like function instead of the usual Euclidean squared distance. Within this interpretation, we derive in a simple way convergence and efficiency estimates. We then propose an Entropic mirror descent algorithm for convex minimization over the unit simplex, with a global efficiency estimate proven to be mildly dependent in the dimension of the problem.     

Positive and negative solutions of a class of nonlocal problems

The existence of nontrivial solutions of Kirchhoff type equations is an important nonlocal quasilinear problem. In this paper, still by using the invariant sets of descent flow, we obtain positive and negative solutions of a class of nonlocal quasilinear elliptic boundary value problems as follows: