Optimality for set functions with values in ordered vector spaces

Let (X, , ) be a finite atomless measure space,L a convex subfamily of , andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:LY beC-convex andG:LZ beD-convex set functions. Consider the following optimization problem (P): minimizeF(), subject to L andG() _D . The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.This work was partially supported by National Science Council, Taipei, Taiwan. This paper was written while the first author was visiting at the University of Iowa, 1987-88.The authors would like to express their gratitude to the two anonymous referees for their valuable comments. Also, they would like to thank Professor P. L. Yu for his encouragement and suggestions which improved the material presented here considerably.     

Approximate first-order and second-order directional derivatives of a marginal function in convex optimization

Given a convex functionf: ^p × ^q (–, +], the marginal function is defined on ^p by (x)=inf{f(x, y)|y ^q }. Our purpose in this paper is to express the approximate first-order and second-order directional derivatives of atx ₀ in terms of those off at (x ₀,y ₀), wherey ₀ is any element for which (x ₀)=f(x ₀,y ₀).The author is indebted to one referee for pointing out an inaccuracy in an earlier version of Theorem 4.1.     

2D层状焦磷酸锰(Ⅱ)[NH4]2[Mn3(P2O7)2(H2O)2]的结构和磁性研究

A two-dimensional layered manganese(Ⅱ) pyrophosphate, [NH₄]₂[Mn₃(P₂O₇)₂(H₂O)₂] (1), has been synthesized hydrothermally. Its structure is determined by single-crystal X-ray diffraction analysis. Its structure is built up by MnO₆ octahedra and P₂O₇ units, with ammonium NH₄⁺ cations residing in the interlayer regions. The manganese pyrophosphate layer consists of infinite chains of cis and trans edge sharing MnO₆ octahedra linked by P₂O₇ units. Magnetic susceptibility measurements show that this compound exhibits ferrimagnetic-like ordering below 3.2 K. Further study shows that there are two possible superexchange pathways via the oxygen anions in Mn²⁺ ions. Owing to the different bridging modes of O^2-, the competition between ferromagnetic interactions and antiferromagnetic interactions in Mn²⁺ ions results in the ferrimagnetic behavior of 1.     

Nanostructural Approach to Proton Ordering in Gas Hydrate Cages

A nanostructural approach to analysis of proton ordering in gas hydrate cages has been worked out within the framework of the topological model of strong and weak H-bonds. The approach involves rejection of the periodic boundary conditions, decomposition of the H-bond net into spherical layers, and two-dimensional drawing of the structure of spherical (spheroidal) fragments in the form of conjugate Schlegel diagrams. To analyze proton ordering in the spherical fragments composed of gas hydrate voids, we used the simulated annealing procedure and the correlation extension method proposed earlier.     

Structure formation in heavy metal oxyhydrates at low rates of gel formation

Oxyhydrate gels have a hydrophilic surface, due to which they undergo destruction and secondary polymerization in aqueous media. Prolonged storage in aqueous solution gives rise to regions with selfsimilar helical ordering in gels. Structuring of this kind is also observed when synthesis is conducted under conditions that provide low gelation rates. Electromagnetic UV and visible radiation is another means to change the gel structure; it makes the oligomer species pass into the excited state, due to which one of the directions of structuring becomes dominant. This work summarizes the results of computer simulation of gel agglomerates. For oxyhydrate systems, helical ordering was found to be one of the local energy minima. The units of a macrohelix can lie at various angles relative to one another, and they can change, after absorption of energy, the helix pitch and the order of elements in the helix.     

Finding normal solutions in piecewise linear programming

Letf: ⁿ (–, ] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg min _x ¦x¦²/2+t f(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)} _t 0, minimizingf over an ellipsoid, and choosing penalty parameters inL ₁QP methods for strictly convex QP.This research was supported by the State Committee for Scientific Research under Grant 8S50502206.     

Potential reduction method for harmonically convex programming

In this paper, we introduce a potential reduction method for harmonically convex programming. We show that, if the objective function and them constraint functions are allk-harmonically convex in the feasible set, then the number of iterations needed to find an -optimal solution is bounded by a polynomial inm, k, and log(1/). The method requires either the optimal objective value of the problem or an upper bound of the harmonic constantk as a working parameter. Moreover, we discuss the relation between the harmonic convexity condition used in this paper and some other convexity and smoothness conditions used in the literature.The authors like to thank Dr. Hans Nieuwenhuis for carefully reading this paper and the anonymous referees for the worthy suggestions.     

Real analysis related to the Monge-Ampère equation

In this paper we consider a family of convex sets in , , , , satisfying certain axioms of affine invariance, and a Borel measure satisfying a doubling condition with respect to the family The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of This is achieved by showing first a Besicovitch-type covering lemma for the family and then using the doubling property of the measure The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to

     

On a class of functions with equal infima over a domain and its boundary

In this paper, we present a class of functions:f:X such that inf _xX f(x)= , whereX is a nonempty, finitely compact and convex set in a vector space andB _x ={xX: y aff(X){x:[x, y]X={x}. Our main tool is a recent minimax theorem by Ricceri (Ref. 1).