Upper Bound Design for the Lipschitz Constant of the F G ( ν , q )-Entropy Operator

This paper develops an upper bound design method of the Lipschitz constant for the generalized Fermi–Dirac information entropy operator with a polyhedral admissible set. We introduce the concept of a normal operator from this class in which the constraint matrix has normalized columns. Next, we establish a connection between the normal and original operator. Finally, we demonstrate that the normal operator is majorized by the linear one and find numerical characteristics of this majorant.


Introduction
Mathematical modeling methods based on constrained optimization of parameterized entropy functions (or functionals) are widely used in different applications such as image recognition in computerized tomography [1,2], dynamic regression models estimation [3], and randomized machine learning [4], to name a few.
Entropy functions and admissible sets depend on basic variables and parameters, some being fixed while others take values from their definitional domains.An entropy operator is an operator that maps the definitional domains of variable parameters into a set of entropy-optimal basic variables.In general form an entropy operator can be written as where H(u, v) denotes an entropy function of basic variables u and parameters v while D gives an admissible set with parameters g.This paper deals with the so-called F G (ν, q)-entropy operators defined by where H F (u, ν) denotes the generalized Fermi-Dirac information entropy, N and Q are m-dimensional unit and r-dimensional parallelepipeds, respectively, and where In what follows, we will suggest an upper bound design method for the local Lipschitz constant (over a compact set) of the entropy operators belonging to this class.
The Lipschitz constant plays an important role in theory of dynamic systems, including for the systems with entropy operator [5], in dynamic procedures of computerized tomography [6], and others.

Problem Statement and Logical Scheme of Solution
Consider the entropy operator F G (ν, q) (2) with the entropy function and the matrix B (3) of full rank r.The definitional domains of the variables have the form with constants σ 1 , . . ., σ m , ν * 1 , . . ., ν * m , and 1 , . . ., r ; q * 1 , . . ., q * r are small fixed values.The local Lipschitz constant of the F G (ν, q)-entropy operator (2) and ( 5) over the set S (6) is a value L F S that satisfies the inequality The problem is to find an upper bound LF S for the Lipschitz constant L F S .The upper bound design method for the Lipschitz constant of the F G (ν, q)-entropy operator involves three main ideas as follows.The first idea is to select a suitable operator from the same class for which (1) there exists a close relation to the original operator and (2) it is simpler to obtain an upper bound for the Lipschitz.Such a suitable operator will be called normal and denoted by F 0 G (ν, q) (Sections 3 and 4).The second idea concerns majorant design: in the beginning, the normal operator F 0 G (ν, q) is majorized by the normal operator B0 (ν, q) (with the Boltzmann entropy) and then an appropriate majorant in form of a linear operator is constructed for it (Section 5).Finally, the third idea deals with the estimation and localization of the eigenvalues of the linear majorant operator (Section 6).

Normal Form of Entropy Operator
The normal entropy operator F 0 G (ν, q) is given by where denotes the generalized Fermi-Dirac information entropy [7]; means an admissible set; • the matrix W has full rank r, normalized columns e (r) , e (m) are unit vectors with r and m-dimensions, respectively, and the dominant diagonal of the matrix WW , i.e., • the definitional domain of the vectors ν, q is with constants σ 1 , . . ., σ m , ν * 1 , . . ., ν * m , and 1 , . . ., r ; q * 1 , . . ., q * r are small fixed values.

4.
Relationship between F G (ν, q) and F 0 G (ν, q) Theorem 1.There exists a matrix P ≥ 0 of dimensions m × r that satisfies the conditions Proof.Consider system (3): Premultiplying this equality by a nondegenerate matrix P of dimensions m × r yields Select the matrix P so that the conditions of Theorem 1 hold.This is a system of r equations with respect to (m × r) variables-the elements of the matrix P. Because the matrix B in ( 16) is nondegenerate, this system has a set of solutions.
For example, choose the solution that maximizes the entropy subject to (14).

5.
Majorants of F 0 G (ν, q)-Entropy Operator Let us use the B 0 (ν, q)-entropy operator as a majorant for the F 0 G (ν, q)-entropy operator, defining some domain S ⊂ S (6) where this can be done: The B 0 (ν, q)-entropy operator has the form where denotes the generalized Boltzmann information entropy.The characteristics of the admissible set are the same as for the F 0 G (ν, q)-entropy operator ( 10)-( 12).The Lagrange function of the B 0 (ν, q)-entropy operator is written as where λ 1 , . . ., λ r indicate Lagrange multipliers.
The first-order optimality condition for this function leads to the following system of equations with respect to the dual variables z = exp(−λ): B(ν, z(ν, q)) = 1. (23) Here the vector B consists of the elements Similar equations hold for the F 0 G (ν, q)-entropy operator.In accordance with (A6), where the vector F consists of the elements Theorem 2. The vector function B(ν, z(ν, q)) (23) and ( 24) is a majorant in the variable z for the function F(ν, z(ν, q)) (25) and (26), i.e., F(ν, z(ν, q)) ≤ B(ν, z(ν, q)), in the domain S (19) described by the parameters Since the terms in the above sums are positive, it suffices that Consequently, This system of inequalities holds if each term in the left-hand side of each inequality is smaller than its counterpart in the right-hand side, i.e., Denote where the variables G min , σ min , ν max , and t max are defined by Equalities (28).Now, get back to the system of Equation (25).This system has a nonzero solution z 0 = {z 0 , . . ., z 0 } (35) if Here we have utilized the property (11) of the normal F 0 G (ν, q)-entropy operator.Then, Expression (36) gives This condition guarantees the non-emptiness of the set Q (29).
For obtaining an appropriate vector z * , let us construct a majorant for the function A(ν, q, z) under fixed ν and q.Using the inequality where from [10], we get the upper bound where c kj are the elements of the matrix C (43).By Theorem 3.1 from [9], the nonzero solutions to Equation (44) can be majorized by the nonzero solutionsto Equation (42), i.e., z(ν, q) z(ν, q) under fixed (ν, q) ∈ S (49)

S
On the strength of Theorems 2 and 3 (also see [8]), we may write the following upper bound for the local Lipschitz constant: where x min = m G min ν min z 0 1 − ν max + ν min z 0 , In these formulas, the variables z 0 and z * are defined by Equalities (40) and (41), respectively.

Conclusions
The method of upper bound design for the Lipschitz constant of the F G (ν, q)-entropy operator is developed.It is based on the normal entropy operator, and the definition of relation between normal and original operators.Then, the linear majorant of the normal operator is defined, and estimation of the Lipschitz constant for the original operator is performed.
The linear majorant method is important for investigation of the properties of entropy operators, for instance, for Boltzmann and Einstein operators [7].The F G (ν, q)-entropy operator is characterized by the parametrical problem for conditional maximization.Also, there exists a wide class of entropy operators that are described by mathematical programming problems.Development of the method of the upper bound design for these operators will represent interesting progress.
Proof.It suffices to check the signs of corresponding derivatives.Consider Equalities (A4).
The derivatives of the function Ω 0 k (ν, λ) with respect to the variables λ are given by (A10) These expressions vanish if at least one of the variables z s = 0.The proof of Lemma A1 is complete.Lemma A2.Under the assumptions of Lemma A1, the systems of Equations (A4) and (A6) determine the explicit functions λ(ν, q) and z(ν, q), respectively.