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Da Lat, Vietnam

Li G.,University of New South Wales | Mordukhovich B.S.,Wayne State University | Nghia T.T.A.,Oakland University | Pham T.S.,University of Dalat
Mathematical Programming | Year: 2016

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish a far-going extension of the Łojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (Hölder type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings. © 2016 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society

Nguyen H.N.,Vietnamese Academy of Forest science | Tran V.T.,University of Dalat
Annales Botanici Fennici | Year: 2014

A clambering bamboo from southern Vietnam is described as a new species, Maclurochloa locbacensis N.H. Nguyen & V.T. Tran (Poaceae, Bambusoideae) and illustrated in line drawings. It is similar to M. montana and M. tonkinensis, but differs by having deeply concave culm sheaths and flat stigmas. © Finnish Zoological and Botanical Publishing Board 2014.

Vui H.H.,Institute of Mathematics | So'n P.T.,University of Dalat
SIAM Journal on Optimization | Year: 2010

This paper studies the representation of a positive polynomial f on a closed semialgebraic set S := {κ ∈ R n | gi(κ) = 0, i = 1, . . . , l, hj(κ) ≥ 0, j = 1, . . . ,m} modulo the so-called critical ideal I(f, S) of f on S. Under a constraint qualification condition, it is demonstrated that, if either f > 0 on S or f ? 0 on S and the critical ideal I(f, S) is radical, then f belongs to the preordering generated by the polynomials h1, . . . , hm modulo the critical ideal I(f, S). These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge monotonically, increasing to the infimum value f ? := inf κ∈S f(κ) of f on S, provided that the infimum value is attained at some point. Besides, we shall construct a finite set in R containing the infimum value f ?. Moreover, some relations between the Fedoryuk [Soviet Math. Dokl., 17 (1976), pp. 486-490] and Malgrange [Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys. 126, Springer, Berlin, 1980, pp. 170-177] conditions and coercivity for polynomials, which are bounded from below on S, are also established. In particular, a sufficient condition for f to attain its infimum on S is derived from these facts. We also show that every polynomial f, which is bounded from below on S, can be approximated in the l1-norm of coefficients by a sequence of polynomials f ∈ that are coercive. Finally, it is shown that almost every linear polynomial function, which is bounded from below on S, attains its infimum on S and has the same asymptotic growth at infinity. © 2010 Society for Industrial and Applied Mathematics.

Dinh S.T.,Institute of Mathematics | Ha H.V.,Institute of Mathematics | Pham T.S.,University of Dalat
Mathematical Programming | Year: 2013

In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let f0 and f1,…,fp:Rn→R be convenient polynomial functions, and let (formula presented.) Under the assumption that the map (formula presented.) is non-degenerate at infinity, we show that if f0 is bounded from below on S, then f0 attains its infimum on S. © 2013, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Tran A.,University of Dalat | Truong T.,University of Dalat | Le B.,Ho Chi Minh City University of Natural Sciences
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

This paper shows a mathematical foundation for almost important features in the problem of discovering knowledge by association rules. The class of frequent itemsets and the association rule set are partitioned into disjoint classes by two equivalence relations based on closures. Thanks to these partitions, efficient parallel algorithms for mining frequent itemsets and association rules can be obtained. Practically, one can mine frequent itemsets as well as association rules just in the classes that users take care of. Then, we obtain structures of each rule class using corresponding order relations. For a given relation, each rule class splits into two subsets of basic and consequence. The basic one contains minimal rules and the consequence one includes in the rules that can be deducted from those minimal rules. In the rest, we consider association rule mining based on order relation min. The explicit form of minimal rules according to that relation is shown. Due to unique representations of frequent itemsets through their generators and corresponding eliminable itemsets, operators for deducting all remaining rules are also suggested. Experimental results show that mining association rules based on relation min is better than the ones based on relations of minmin and minMax in terms of reduction in mining times as well as number of basic rules. © 2012 Springer-Verlag.

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