Fenchel's duality theorem in generalized geometric programming

Fenchel's duality theorem is extended to generalized geometric programming with explicit constraints—an extension that also generalizes and strengthens Slater's version of the Kuhn-Tucker theorem.This research was sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-73-2516.     

A new geometric condition for Fenchel's duality in infinite dimensional spaces

In 1951, Fenchel discovered a special duality, which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions, using conjugates. Fenchel's duality is central to the study of constrained optimization. It requires an existence of an interior point of a convex set which often has empty interior in optimization applications. The well known relaxations of this requirement in the literature are again weaker forms of the interior point condition. Avoiding an interior point condition in duality has so far been a difficult problem. However, a non-interior point type condition is essential for the application of Fenchel's duality to optimization. In this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions. We also establish a necessary and sufficient condition for the ε-subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions. Our results offer further insight into Fenchel's duality. Dedicated to Terry Rockafellar on his 70th birthday     

On Rosenthal-Shields' problem

In this paper it will be shown that the spectrum of every unicellular unilateral weighted shift operator on a symmetric Banach space is the singleton set {0}. From this, we give an affirmative answer to Rosenthal-Shields' problem. Project supported by the National Natural Science Foundation of China     

关于Thompson问题

1987年本文第二作者给J.G.Thompson教授的信中提出猜想:所有有限单群S,都能够用S的阶和S的元的阶的集合统一的刻画.在回信中,Thompson提出了他的关于有限群G可解性判断的问题.当G的素图非连通时,本文给出了J.G.Thompson问题的肯定回答.     

On vector equilibrium problem

This paper presents some existence results of a vector equilibrium problem. The several important special cases of the vector equilibrium problem are also discussed.     

On single courier problem

A new set of NP problems defined as Courier Problems that is motivated from the requirements in railway wagon scheduling is proposed. The general version includes many mobile couriers. The simplest version of this will consider a single courier. An algorithm to transform the single courier problem into a traveling salesman problem is presented.     

On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

On the cancellation problem

Let k be an algebraically closed field. For every n ≥ 8 we give examples of Zariski open, dense, affine subsets of the affine space A ⁿ (k) which do not have the cancellation property. Dedicated to Professor Mikhail Zaidenberg. The author was partially supported by the grant of Polish Ministry of Science, 2006–2009.     

On a Quasi-Ramsey problem

It is proved that if a graph G has atleast cn log n vertices, then either G or its complement G contains a subgraph H with atleast n vertices and minimum degree atleast | V(H)|/2. This result is not far from being best possible, as is shown by a rather unusual random construction. Some related questions are also discussed.     

On the Dirichlet problem

A particular case of the Dirichlet problem is solved using the Convergence Theorem for discrete-time martingales and the mean value property of harmonic functions as the main tools.     

On Zariski's multiplicity problem

We show that to answer affirmatively Zariski's question concerning the topological invariance of the multiplicity of complex analytic hypersurfaces at isolated singular points, it suffices to prove two combined statements, each of which may be obtained separately.