Multiobjective problems of convex geometry

Abstract

UDC 514.172:517.982:519.81:519.855 Abstract: Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body x, we try to maximize the volume of x and minimize the width of x simultaneously. These problems are addressed along the lines of multiple criteria decision making. We describe the Pareto-optimal solutions of isoperimetric-type vector optimization problems on using the techniques of the space of convex sets, linear majorization, and mixed volumes. The birth of the theory of extremal problems is usually tied with the mythical Phoenician Princess Dido. Virgil told about the escape of Dido from her treacherous brother in the first chapter of The Aeneid. Dido had to decide about the choice of a tract of land near the future city of Carthage, while satisfying the famous constraint of selecting "a space of ground, which (Byrsa call'd, from the bull's hide) they first inclos'd." By the legend, Phoenicians cut the oxhide into thin strips and enclosed a large expanse. Now it is customary to think that the decision by Dido was reduced to the isoperimetric problem of finding a figure of greatest area among those surrounded by a curve whose length is given. It is not excluded that Dido and her subjects solved the practical versions of the problem when the tower was to be located at the sea coast and part of the boundary coastline of the tract was somehow prescribed in advance. The foundation of Carthage is usually dated to the ninth century bce when there was no hint of the Euclidean geometry, the cadastral surveying was the job of harpedonaptae, and measuring the tracts of land was used in decision making. Rope-stretching around stakes leads to convex figures. The Dido problem has a unique solution in the class of convex figures provided that the fixed nonempty part of the boundary is a convex polygonal line. Decision making has become a science in the twentieth century. The presence of many contradictory conditions and conflicting interests is the main particularity of the social situations under control of today. Management by objectives is an exceptional instance of the stock of rather complicated humanitarian problems of goal agreement which has no candidates for a unique solution. The extremal problems of optimizing several parameters simultaneously are collected nowadays un- der the auspices of vector or multiobjective optimization. Search for control in these circumstances is multiple criteria decision making. The mathematical apparatus of these areas of research is not rather sophisticated at present (see (1, 2) and the references therein). The today's research deals mostly with the concept of Pareto optimality (e.g., (3-6)). Let us explain this approach by the example of a bunch of economic agents each of which intends to maximize his own income. The Pareto efficiency principle asserts that as an effective agreement of the conflicting goals it is reasonable to take any state in which nobody can increase his income in any way other than diminishing the income of at least one of the other fellow members. Formally speaking, this implies the search of maximal elements of the set comprising the tuples of incomes of the agents at every state; i.e., some vectors of a finite-dimensional arithmetic space endowed with the coordinatewise order. Clearly, the concept of Pareto optimality was already abstracted to arbitrary ordered vector spaces (for more detail see (7-10)).