拟微分的一个性质

不可微优化的理论研究是从研究凸函数开始的,人们首先发现,对于R~n上有限凸函数f,,凸紧集,使利用次梯度的概念,得到了很多凸函数的重要性质及最优性条件,并构造了一些次梯度算法。 1980年左右,V.F.Demyanov在研究极大极小问题的过程中提出了拟可微的概念[1]. 定义设f是定义于某一非空开集上的实函数,如果是     

整凸规划的某些理论

本文讨论目标函数和约束函数皆为凸函数的整规划问题，首先利用精确罚函数把整凸规划求解化为求凸函数极小整解问题，还讨论了凸函数极小整解的最优性条件。     

一类广义弧式凸函数的局部极小与整体极小的关系

1 引言凸函数有一重要性质,即局部极小必为整体极小．在对凸函数所做的各种推广中,关于局部极小与整体极小的关系问题讨论甚多．本文引进了一种新的广义凸函数,即所谓的弧式严格局部拟凸函数,并证明了一个定义在Rn中紧弧式连通集M上的连续函数．若其每个局部极小均为整体极小,则此函数必为弧式严格局部拟凸函数．这是作者见     

关于多元多项式凸逼近

§1.引言 在多项式保形逼近理论中面临以下两个基本问题: 问题1.对于k≥2,R~k中是否存在k维紧凸集E及C(E)上的保凸正线性算子列L_n:C(E)→P_n满足:?凸函数f∈C(E),||L_nf-f||_E→0(n→∞)? 问题2.对于k≥2以及R~k中的任意k维紧凸集E和任意凸函数f∈C(E),是否存在一列多项式p_n∈P_n,使每一个p_n在E上为凸函数,并且||p_n-f||_E→0(n→∞)?     

非光滑(F,ρ,)θ-d一致不变凸广义分式规划的对偶性

结合F-凸,η-不变凸及d一致不变凸的概念给出了非光滑广义(F,ρ,θ)-d一致不变凸函数;就一类在凸集C上目标函数为Lipschitz连续的带有可微不等式约束的广义分式规划,提出一个对偶,并利用在广义Kuhn-Tucker约束品性或广义Arrow-Hurwicz-Uzawa约束品性的条件下得到的最优性必要条件,证明相应的弱对偶定理、强对偶定理及严格逆对偶定理.     

紧－凸性与紧－光滑性

本文首先通过暴露集和暴露泛函的概念引入了闭凸集的紧－严格凸、紧－强凸、紧－一致凸及紧－非常凸等概念。并用对偶映射给出了Ｂａｎａｃｈ空间的两种新光滑性—紧－一致光滑与紧－非常光滑。然后特别研究了Ｂａｎａｃｈ空间的紧－非常凸与紧－非常光滑。此外还得到关于对偶映射的两个新结果。     

紧—凸性与紧—光滑性

本文首先通过暴露集和暴露泛函的概念引入卫闭凸集的紧－严格凸、紧－强凸、紧－一致凸及紧－非常凸等概念。用对偶映射给出了Ｂａｎａｃｈ空间的两种新光滑性－紧－一致光滑与紧－非常光滑。然后特别研究了Ｂａｎａｃｈ空间的紧－非常凸与紧－非常光滑。此外还得到关于对偶映射的两个新结果。     

(γ,α)型广义强凸性

引入(γ,α)型广义强凸集与强凸函数,讨论了广义强凸性质,并在此基础上提出对强凸函数进行分类的标准和判定方法.然后引入标准强凸函数概念,推出最小标准强凸函数形式,并探讨了广义强凸集与强凸函数的关系.     

无限维空间中强拟凸向量优化问题有效解集的连通性

本文在无穷维空间引进增强凸变换的概念，在约束集是紧和凸的条件下目标函数是连续和强拟凸的，我们得取到向量极小问题有效解集的连通性。     

一类新的弧式连通函数

凸分析是非光滑分析中发展比较成熟的一个方向,关于凸集、凸函数理论的奠基工作可以追溯到本世纪初的Jensen和Minkowski的著作。然而真正引起人们重视的是Von.Neumann.Dantzing,Kuhn,Tucker等人对运筹学、规划论的研究.Rockaffellar的“Convex Analysis”一书使凸分析成为一数学分科。但是实际抽象出来的许多实际问题     

A criterion for checking if a convex set belongs to the union of a finite number of convex sets

A necessary and sufficient condition for the inclusion of a convex compact set in the union of a finite number of convex sets is proved. This condition obtained using the convex analysis techniques is a condition on the maximin of a given function. Using the dynamic programming, checking this condition is reduced to evaluating a set of functions and checking a condition for their values. The reduced form of the criterion is more convenient from the computational point of view.     

Piecewise-linear programming: The compact (CPLP) algorithm

A compact algorithm is presented for solving the convex piecewise-linear-programming problem, formulated by means of a separable convex piecewise-linear objective function (to be minimized) and a set of linear constraints. This algorithm consists of a finite sequence of cycles, derived from the simplex method, characteritic of linear programming, and the line search, characteristic of nonlinear programming. Both the required storage and amount of calculation are reduced with respect to the usual approach, based on a linear-programming formulation with an expanded tableau. The tableau dimensions arem×(n+1), wherem is the number of constraints andn the number of the (original) structural variables, and they do not increase with the number of breakpoints of the piecewise-linear terms constituting the objective function.     

On the stability of inner and outer approximations of a convex compact set by a ball

The finite-dimensional problems of outer and inner estimation of a convex compact set by a ball of some norm (circumscribed and inscribed ball problems) are considered. The stability of the solution with respect to the error in the specification of the estimated compact set is generally characterized. A new solution criterion for the outer estimation problem is obtained that relates the latter to the inner estimation problem for the lower Lebesgue set of the distance function to the most distant point of the estimated compact set. A quantitative estimate for the stability of the center of an inscribed ball is given under the additional assumption that the compact set is strongly convex. Assuming that the used norm is strongly quasi-convex, a quantitative stability estimate is obtained for the center of a circumscribed ball.     

The infima of functionals of a special kind on a compact convex set

It is proved that the infimum of the ratio of a concave functional and a convex functional is attained at the extreme points of a compact convex set in a normed linear space. A criterion for the membership of a given element in the set of extreme points is proposed and the existence of a strongly convex functional on a compact set is shown.     

Proximal Methods in View of Interior-Point Strategies

This paper deals with regularized penalty-barrier methods for convex programming problems. In the spirit of an iterative proximal regularization approach, an interior-point method is constructed, in which at each step a strongly convex function has to be minimized and the prox-term can be scaled by a variable scaling factor. The convergence of the method is studied for an axiomatically given class of barrier functions. According to the results, a wide class of barrier functions (in particular, logarithmic and exponential functions) can be applied to design special algorithms. For the method with a logarithmic barrier, the rate of convergence is investigated and assumptions that ensure linear convergence are given.     

Numerical Algorithm for Minimizing a Convex Function on the Intersection of a Smooth Surface and a Convex Compact Set

Computational Mathematics and Mathematical Physics - A numerical algorithm for minimizing a convex function on the set-theoretic intersection of a smooth surface and a convex compact set in...     

On Well Definedness of the Central Path

We study the well definedness of the central path for a linearly constrained convex programming problem with smooth objective function. We prove that, under standard assumptions, existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given. We show that, under an additional assumption on the objective function, the central path converges to the analytic center of the optimal set.     

Iterative algorithm for minimizing a convex function at the intersection of a spherical surface and a convex compact set

A numerical algorithm for minimizing a convex function on the set-theoretic intersection of a spherical surface and a convex compact set is proposed. The idea behind the algorithm is to reduce the original minimization problem to a sequence of convex programming problems. Necessary extremum conditions are examined, and the convergence of the algorithm is analyzed.     

Iteration-complexity of first-order penalty methods for convex programming

This paper considers a special but broad class of convex programming problems whose feasible region is a simple compact convex set intersected with the inverse image of a closed convex cone under an affine transformation. It studies the computational complexity of quadratic penalty based methods for solving the above class of problems. An iteration of these methods, which is simply an iteration of Nesterov’s optimal method (or one of its variants) for approximately solving a smooth penalization subproblem, consists of one or two projections onto the simple convex set. Iteration-complexity bounds expressed in terms of the latter type of iterations are derived for two quadratic penalty based variants, namely: one which applies the quadratic penalty method directly to the original problem and another one which applies the latter method to a perturbation of the original problem obtained by adding a small quadratic term to its objective function.     

An algorithm for constrained convex optimization

This paper presents a feasible direction algorithm for the minimization of a pseudoconvex function over a smooth, compact, convex set. We establish that each cluster point of the generated sequence is an optimal solution of the problem without introducing anti-jamming procedures. Each iteration of the algorithm involves as subproblems only one line search for a zero of a continuously differentiable convex function and one univariate function minimization on a compact interval.