How to divide a territory? A new simple differential formalism for optimization of set functions

In many practical problems, we must optimize a set function, i.e., nd a set A for which f(A) ! max, where f is a function deened on the class of sets. Such problems appear in design, in image processing, in game theory, etc. Most optimization problems can be solved (or at least simpliied) by using the fact that small deviations from an optimal solution can only decrease the value of the objective function; as a result, some derivative must be equal to 0. This approach has been successfully used, e.g., for set functions in which the desired set A is a shape, i.e., a smooth (or piece-wise smooth) surface. In some real-life problems, in particular, in the territorial division problem, the existing methods are not directly applicable. For such problems, we design a new simple diierential formalism for optimizing set functions.

[1]  R. Fisher.  The Advanced Theory of Statistics , 1943, Nature.

[2]  A. Rosenthal,et al.  What are Set Functions , 1948 .

[3]  E. M. Lifshitz,et al.  Classical theory of fields , 1952 .

[4]  S. Vajda,et al.  GAMES AND DECISIONS; INTRODUCTION AND CRITICAL SURVEY. , 1958 .

[5]  E. Hewitt,et al.  Theory of functions of a real variable , 1960 .

[6]  R. A. Silverman,et al.  Integral, Measure and Derivative: A Unified Approach , 1967 .

[7]  Charles A. Hayes,et al.  Derivation and Martingales , 1970, Ergebnisse der Mathematik und ihrer Grenzgebiete.

[8]  R. Leighton,et al.  Feynman Lectures on Physics , 1971 .

[9]  Curtis F. Gerald,et al.  APPLIED NUMERICAL ANALYSIS , 1972, The Mathematical Gazette.

[10]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[11]  T. Bukowski,et al.  Integral. , 2019, Healthcare protection management.

[12]  Bruce L. Golden,et al.  Optimisation , 1982, IEEE Trans. Syst. Man Cybern..

[13]  Kensuke Tanaka,et al.  The Multiobjective Optimization Problem of Set Function , 1984 .

[14]  J. Hartigan.  Estimation of a Convex Density Contour in Two Dimensions , 1987 .

[15]  Michael Gelfond,et al.  Theory of deductive systems and its applications , 1987 .

[16]  M. Delfour,et al.  Shape sensitivity analysis via a penalization method , 1988 .

[17]  D. Nolan.  The excess-mass ellipsoid , 1991 .

[18]  S. Vavasis.  Nonlinear optimization: complexity issues , 1991 .

[19]  G. Sawitzki,et al.  Using excess mass estimates to investigate the modality of a distribution , 1991 .

[20]  Jan Sokolowski,et al.  Introduction to shape optimization , 1992 .

[21]  P. Pardalos.  Complexity in numerical optimization , 1993 .

[22]  L. Doyen.  Filippov and invariance theorems for mutational inclusions of tubes , 1993 .

[23]  Juliette Mattioli,et al.  Differential inclusions for mathematical morphology , 1993, Optics & Photonics.

[24]  M. Delfour,et al.  Shape Analysis via Oriented Distance Functions , 1994 .

[25]  L. Doyen.  Inverse Function Theorems and Shape Optimization , 1994 .

[26]  W. Polonik.  Measuring Mass Concentrations and Estimating Density Contour Clusters-An Excess Mass Approach , 1995 .

[27]  J. Mattioli.  Minkowski operations and vector spaces , 1995 .

[28]  Michel C. Delfour,et al.  A Boundary Differential Equation for Thin Shells , 1995 .

[29]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[30]  Hung T. Nguyen,et al.  A First Course in Fuzzy Logic , 1996 .

[31]  Vladik Kreinovich,et al.  Astrogeometry, error estimation, and other applications of set-valued analysis , 1996, SGNM.

[32]  M. Delfour,et al.  Tangential Differential Equations for Dynamical Thin/Shallow Shells , 1996 .

[33]  V. Kreinovich,et al.  Problems of reducing the exhaustive search , 1996 .

[34]  Vladik Kreinovich,et al.  Astrogeometry: Towards Mathematical Foundations , 1996 .

[35]  andVladik Kreinovichc.  Astrogeometry: Geometry Explains Shapes of Celestial Bodies 1. Our Main Idea: Astroshapes as Geometric Objects 1.1. Physical Background , 1996 .

[36]  J. M. Gutiérrez,et al.  NON-EQUILIBRIUM THERMODYNAMICS EXPLAINS SEMIOTIC SHAPES : APPLICATIONS TO ASTRONOMY AND TO NON-DESTRUCTIVE TESTING OF AEROSPACE SYSTEMS , 1997 .

[37]  V. Kreinovich.  Computational Complexity and Feasibility of Data Processing and Interval Computations , 1997 .

[38]  Vladik Kreinovich,et al.  Computational Geometry and Artificial Neural Networks: A Hybrid Approach to Optimal Sensor Placement for Aerospace NDE , 1997 .

[39]  Olaf Wolkenhauer,et al.  Random-sets: theory and applications , 2001 .