A new technique to derive tight convex underestimators (sometimes envelopes)

M. Locatelli ()
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M. Locatelli: Università di Parma

Computational Optimization and Applications, 2024, vol. 87, issue 2, No 6, 475-499

Abstract: Abstract The convex envelope value for a given function f over a region X at some point $$\textbf{x}\in X$$ x ∈ X can be derived by searching for the largest value at that point among affine underestimators of f over X. This can be computed by solving a maximin problem, whose exact computation, however, may be a hard task. In this paper we show that by relaxation of the inner minimization problem, duality, and, in particular, by an enlargement of the class of underestimators (thus, not only affine ones) an easier derivation of good convex understimating functions, which can also be proved to be convex envelopes in some cases, is possible. The proposed approach is mainly applied to the derivation of convex underestimators (in fact, in some cases, convex envelopes) in the quadratic case. However, some results are also presented for polynomial, ratio of polynomials, and some other separable functions over regions defined by similarly defined separable functions.

Keywords: Convex underestimators; Convex envelopes; Quadratic programming; Polynomial programming; Separable functions (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-023-00534-8

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