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Step-Affine Functions, Halfspaces, and Separation of Convex Sets with Applications to Convex Optimization Problems

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Abstract

We introduce the class of step-affine functions defined on a real vector space and establish the duality between step-affine functions and halfspaces, i.e., convex sets whose complements are convex as well. Using this duality, we prove that two convex sets are disjoint if and only if they are separated by some step-affine function. This criterion is actually the analytic version of the Kakutani–Tukey criterion of the separation of disjoint convex sets by halfspaces. As applications of these results, we derive a minimality criterion for solutions of convex vector optimization problems considered in real vector spaces without topology and an optimality criterion for admissible points in classical convex programming problems not satisfying the Slater regularity condition.

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Notes

  1. A partial order (i.e., a reflexive, transitive, and antisymmetric binary relation) \(\preccurlyeq\) defined on a set \(Z\) is called a total (or linear) order if, for any \(z_{1},z_{2}\in Z\), we have either \(z_{1}\preccurlyeq z_{2}\) or \(z_{2}\preccurlyeq z_{1}\).

  2. Totally ordered sets \((U,\preccurlyeq_{U})\) and \((V,\preccurlyeq_{V})\) are said to have the same order type if there exists an order-preserving bijection between them. Two totally ordered sets with the same order type are equipotent. Moreover, two finite totally ordered sets have the same order type if and only if they are equipotent, and their order type is identified with the number of their elements. Infinite totally ordered sets can be equipotent and have different order types. For example, the set of positive integers \({\mathbb{N}}\), the set of integers \({\mathbb{Z}}\), and the set of rational numbers \({\mathbb{Q}}\) have the same cardinality \(\aleph_{0}\), but their order types are different.

  3. A preorder (i.e., a reflexive and transitive binary relation) \(\preceq\) defined on a vector space \(X\) is said to be compatible with the algebraic operations of \(X\) if \(x\preceq y\Longrightarrow x+z\preceq y+z\ \ \ \forall\,x,y,z\in X\) and \(x\preceq y\Longrightarrow\lambda x\preceq\lambda y\ \ \ \forall\,x,y\in X,\ \lambda>0.\) If \(\preceq\) is a compatible preorder, then \(x\preceq y\Longleftrightarrow y-x\in P_{\preceq},\) where \(P_{\preceq}:=\{x\in X\mid 0\preceq x\}\) is the (convex) positive cone of the relation \(\preceq\).

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Funding

This work was supported by the National Program for Scientific Research of the Republic of Belarus for 2016–2020 “Convergence 2020” (project no. 1.4.01).

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, 220072, Minsk, Belarus

    V. V. Gorokhovik

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  1. V. V. Gorokhovik
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Correspondence to V. V. Gorokhovik.

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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 26, No. 1, pp. 51 - 70, 2020 https://doi.org/10.21538/0134-4889-2020-26-1-51-70.

Translated by E. Vasil’eva

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Gorokhovik, V.V. Step-Affine Functions, Halfspaces, and Separation of Convex Sets with Applications to Convex Optimization Problems. Proc. Steklov Inst. Math. 313 (Suppl 1), S83–S99 (2021). https://doi.org/10.1134/S008154382103010X

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