Analysis is a very old term of science with a long history stemming from Ancient Hellas. Morris Kline (1908–1992) stated that the term was introduced by Theon of Alexandria (355–405 CE). Francois Viete (1540–1603) used the term “analytic art” for algebra in 1591 in In Artem Analyticem Isagoge. David Hilbert wrote that analysis is “the most aesthetic and delicately erected structure of mathematics” and called it “a symphony of the infinite.”
Prevalence of one magnitude over the other is one of the earliest abstractions of humankind. In the modern mathematical parlance, the idea of transitive antisymmetric relation had preceded the concept of order.
Order and analysis were combined in the first third of the twentieth century which marked an important twist in the content of mathematics. Mathematical ideas imbued the humanitarian sphere and, primarily, politics, sociology, and economics. Social events are principally volatile and possess a high degree of uncertainty. Economic processes utilize a wide range of the admissible ways of production, organization, and management. The nature of nonunicity in economics transpires: The genuine interests of human beings cannot fail to be contradictory. The unique solution is an oxymoron in any nontrivial problem of economics which refers to the distribution of goods between a few agents. It is not by chance that the social sciences and instances of humanitarian mentality invoke the numerous hypotheses of the best organization of production and consumption, the most just and equitable social structure, the codices of rational behavior and moral conduct, etc.
The twentieth century became the age of freedom. Plurality and unicity were confronted as collectivism and individualism. Many particular phenomena of life and culture reflect their distinction. The dissolution of monarchism and tyranny was accompanied by the rise of parliamentarism and democracy. Quantum mechanics and Heisenberg's uncertainty incorporated plurality in physics. The waves of modernism in poetry and artistry should be also listed. Mankind had changed all valleys of residence and dream.
In mathematics the quest for plurality led to the abandonment of the overwhelming pressure of unicity and categoricity. The latter ideas were practically absent, at least minor, in Ancient Greece and sprang to life in the epoch of absolutism and christianity. Georg Cantor (1845–1918) was a harbinger of mighty changes, claiming that "Wesen der Mathematik liegt gerade in ihrer Freiheit."
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 under 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]-[3] and the references therein).
The today's research deals mostly with the concept of Pareto optimality. Consider 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.
The variational principles of mechanics, precursors of variational calculus, served at least partly to justifying the Christian belief in the unicity and beauty of the act of creation. The extremal problems, generously populating all branches of mathematics, use only scalar targets. Problems with many objectives have become the topic of research rather recently and noticeably beyond mathematics, which explains the substantial gap between the levels of complexity and power of the mathematical tools available for single objective and multiple objective problems. This challenges the task of enriching the stock of vector optimization problems within the theoretical core of mathematics.
For the sake of simplicity, it stands to reason to start with the problems using the concept of Pareto optimality. The point is that each problem of the sort is in fact equivalent to a parametric family of single objective problems that can be inspected by the classical methods. For instance, there is a curve joining the Legendre and Chebyshev polynomials which consists of the polynomials “Pareto-optimal” with respect to the uniform and mean square metrics. Clearly, some physical processes admit description in terms of vector optimization. For instance, we may treat the Leidenfrost effect of evaporation of a liquid drop in the spheroidal state as the problem of simultaneous minimization of the surface area and the width of a drop of a given volume.
Under discussion in this talk will be some discrete dynamic problems of decision making [4] as well as the class of geometrically meaningful vector optimization problems whose solutions can be found explicitly to some extend in terms of conditions on surface area measures. As model examples we give explicit solutions of the Urysohn-type problems aggravated by the flattening condition or the requirement to optimize the convex hull of a few figures. Technically speaking, everything reduces to the parametric programming of isoperimetric type problems with many subsidiary constraints along the lines of the approach developed in [5]. We will pay a special attention to vector problems of the space of Minkowski balls as presented in [6].
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