We show that a compact feasible set of a standard semi-infinite optimization problem can be approximated arbitrarily well by a level set of a single smooth function with certain regularity properties. This function is constructed as the mollification of the lower level optimal value function. Moreover, we use correspondences between KarushKuhn-Tucker points of the original and the smoothed prob...

We consider the global minimization of a multivariate polynomial on a semi-algebraic set defined with polynomial inequalities. We then compare two hierarchies of relaxations, namely, LP relaxations based on products of the original constraints, in the spirit of the RLT procedure of Sherali and Adams (1990), and recent semidefinite programming (SDP) relaxations introduced by the author. The comp...

We investigate the sufficient condition under which each positive b-weakly compact operator is Dunford-Pettis. We also investigate the necessary condition on which each positive b-weakly compact operator is Dunford-Pettis. Necessary condition on which each positive b-weakly compact operator is weakly compact is also considered. We give the operator that is semi-compact, but it is not bweakly. W...

We study the asymptotic behavior of solutions to the second-order evolution equation p(t)u ′′ (t) + r(t)u ′ (t) ∈ Au(t) a.e. t ∈ (0,+∞), u(0) = u0, sup t≥0|u(t)| < +∞, where A is a maximal monotone operator in a real Hilbert space H with A−1(0) nonempty, and p(t) and r(t) are real-valued functions with appropriate conditions that guarantee the existence of a solution. We prove a weak ergodic th...

Let (X, z) be a topological space, and (Y, d) a metric space. A mapping F: X -+ 2’ which associates with every x E X a non-empty subset F(x) OF Y is said to be lower semi-continuous (I.s.c.) (respectively, upper semi-continuous (u.s.c.)) if, for each open set % in Y, the set (X E X: F(x) n %f # @ > (respectively, the set (X E X: F(x) c @ )) is open in X A mapping f: X-a Y is a selection for F i...

Given any maximal monotone operator A : D(A) ⊂ H → 2 in a real Hilbert space H with A−1(0) 6= ∅, it is shown that the sequence of proximal iterates xn+1 = (I + γnA)(λnu+ (1− λn)xn + en) converges strongly to the metric projection of u on A−1(0) for (en) bounded, λn ∈ (0, 1) with λn → 1 and γn > 0 with γn → ∞ as n → ∞. In comparison with our previous paper [Optim. Lett. 4 (2010), 635-641], where...

In this note we show that in a certain subfamily of the Kuranishi family of any half Inoue surface the algebraic dimensions of the fibers jump downwards at special points of the parameter space showing that the upper semi-continuity of algebraic dimensions in any sense does not hold in general for families of compact non-Kähler manifolds. In the Kähler case, the upper semi-continuity always hol...

An S-space is a normal topological space in which each covering by open sets has a refinement which is star-finite, that is, each set of the refinement meets only a finite number of sets of the refinement. Thus a compact ( = bicompact) space is an S-space, and an S-space is paracompact [ l ] . 1 In this note we discuss cartesian products in which one of the factors is an S-space. We show that i...