We analyze minimization algorithms for L\-convex functions in discrete convex analysis, and establish exact bounds for the number of iterations required by the steepest descent algorithm and its variants.

Let (Ω,Σ) be a measurable space and let X be a Banach space. Throughout this paper G will be a countably additive vector measure G : Σ → X. Consider the space L1(G) of (classes of real) G-integrable functions, following the definition of Bartle, Dunford and Schwartz [1] and Lewis [9]. The properties of this space have been studied by Kluvánek and Knowles [7], Okada [10] and Curbera [2]. In the ...

This article is dedicated to Jean Jacques Moreau for his 80th birthday. He is a master of convex analysis. When dealing with differential inclusions of the form Du (x) ∈ E, a.e. in Ω together with some boundary data u = φ on ∂Ω, one is lead to consider several types of convex hulls of sets. It is the aim of the present article to discuss these matters.

Nonsmooth analysis of a broad class of functions taking the form F (x) = mini fi(x), where each fi is a convex function. One element of this class of functions is the distance function, which measures the distance between a point and the nearest point on the nearest obstacle. Many motion planning algorithms are based on the distance function, and thus rigorous analysis of the distance function ...

A SUMMARY BY DIMITRI P. BERTSEKAS We provide a summary of theoretical concepts and results relating to convex analysis, convex optimization, and duality theory. In particular, we list the relevant definitions and propositions (without proofs) of the author’s book “Convex Optimization Theory,” Athena Scientific, 2009. For ease of use, the chapter, section, definition, and proposition numbers of ...

In discrete convex analysis, the scaling and proximity properties for the class of L-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n ≤ 2, while a proximity theorem can be established for any n, but o...

In this short note, we define parametric convex programming (PCP) in a slightly different manner than it is usually done by extending convexity not only to variables but also to the parameters, and we show that the widely applied model predictive control (MPC) technique is a particular case of PCP. The main result of the note is an answer to the inverse question of PCP: which feedback laws can ...

Which convex subsets of C are the numerical range W (A) of some matrix A? This paper gives a precise characterization of these sets. In addition to this we show that for any A there exists a symmetric B of the same size such that W (A) = W (B) thereby settling an open question from [2]. Mathematics Subject Classification (2000). Primary 47A12.