On the continuity of the minimum set of a continuous function

Abstract

PREFACE This Memorandum reports on recent work which is part of a continuing RAND Corporation study of the mathematical and computational aspects of chemical equilibrium theory (see, for example, [4-13]). It may also be regarded as representing part of RAND's work in mathematical programming (see bibliography in [3] ) because it deals with a subject of more general mathematical interest than many of the previous RAND publications in this series. RAND's research in chemical equilibrium theory is complementary to its continuing research on the application of mathematics and computer technology to biochemistry and human physiology [14-21], and is also applicable to, for example, studies of the atmospheres of the earth and other planets, to the computation of the characteristics of propulsive fluids, and to the examination of certain reentry problems. -V-ABSTRACT This Memorandum obtains necessary and sufficient conditions so that the solution of a constrained minimization problem will vary continuously when the constraints and objective function are varied. It also obtains special results in the case that the constraints are linear inequalities.

Diffusion Approximation for an Input-queued Switch Operating under a Maximum Weight Matching Policy

“…Proof. The first property follows from Corollary II.3.1 and Corollary I.3.4 of Dantzig et al [6]. The second property follows by the same argument as in the proof of the second property in Proposition 4.1 of [12].…”

Adaptive and robust radiation therapy optimization for lung cancer

“…Note that when all M defining PMFs are set to p * , problem (4.8) is equivalent to the nominal problem with respect to p * . The proof of Proposition 1 uses Theorem II.2.2 of Dantzig et al [1967] and the fact that the affine functions that define the H(q 1,i , . .…”

“…Suppose that V is a subset of R n and φ : R n → R is a function. Following Dantzig et al [1967], we define the set…”

“…Following Dantzig et al [1967], we say that a sequence of subsets (A n ) ∞ n=1 of a metric space X converges to a set A if lim n→∞ A n = lim n→∞ A n = A. This is the form of set convergence used in Proposition 1.…”

The fundamental continuity theory of optimization on a compact space. I

“…•13-Historical Note : An earlier study of the continuity of the set of optimal solutions is [Dantzig, Folkman and Shapiro, 1967]. It works with feasible regions lying in a metric space and with objective functions coinciding with real-valued incentive functions, and takes the "parameter space" X" as singleton, investigating the continuity of the set of optimal solutions as a function of (i) objective function used and (ii) feasible region.…”