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Convex extensions on combinatorial configurations
Theory and optimization methods
Pliki do pobrania
The present monograph develops a systematic mathematical theory at the interface of combinatorial optimization, convex analysis, and Euclidean geometry. Its central subject is the behaviour of functions defined on finite point configurations embedded in Euclidean space, and the exploitation of this behaviour for the solution of combinatorial optimization problems.
By choosing the Euclidean immersion so that the image of the combinatorial feasible set coincides with the vertex set of its convex hull, a discrete optimization problem is transformed into a continuous one without any loss of information. This structural property is both the condition for the existence of convex extensions and the ultimate source of efficiency of all optimization methods developed in the monograph.
The theoretical core of the work is a complete theory of convex extensions of functions defined on vertex-located configurations: existence theorems for convex, strongly convex, and differentiable extensions; constructive schemes yielding explicit analytic formulas; and a hierarchy of computable extremal estimates and optimality conditions. This theory is applied to the principal classes of finite point configurations arising as Euclidean images of combinatorial sets, including permutation, multipermutation, partial permutation, and Boolean configurations, whose associated polytopes are characterized in full. Building on this foundation, the monograph develops optimization methods including polyhedral and spherical relaxations, branch-and-bound schemes, and penalty function methods, each grounded in the specific geometry of the combinatorial feasible set rather than in general-purpose algorithmic principles.
The book is addressed to researchers and graduate students in combinatorial optimization, convex analysis, and discrete mathematics, as well as to practitioners working on optimization problems in scheduling, assignment, logistics, and engineering design.
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