c Allerton Press, Inc., 2010.
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2010, Vol. 54, No. 10, pp. 75–78. c N.S. Rozinova, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 10, pp. 87–91.
Original Russian Text Optimality Conditions in the Problem of Maximization
of the Difference of Two Convex Functions
N. S. Rozinova1*
(Submitted by V.A. Srochko)
1
Irkutsk State University, ul K. Marksa 1, Irkutsk, 664003 Russia
Received March 18, 2010
Abstract—We consider a quadratic d. c. optimization problem on a convex set. The objective
function is represented as the difference of two convex functions. By reducing the problem to the
equivalent concave programming problem we prove a sufficient optimality condition in the form
of an inequality for the directional derivative of the objective function at admissible points of the
corresponding level surface.
DOI: 10.3103/S1066369X10100117
Key words and phrases: d. c.-maximization problem, necessary and sufficient optimality
conditions.
1. INTRODUCTION
D. c.-optimization problems (d. c. is the difference of two convex functions) have a sufficiently high
level of generality and form a special class of nonconvex structures [1–3]. The first level of investigation
of such problems is connected with establishing the optimality conditions which take into account the
specificity of the problem and extract the set of extremal points. It is interesting that results obtained in
this area represent not only the necessary global optimality conditions, but also sufficient ones; earlier
this property was observed only in convex problems.
In this paper we consider a quadratic d. c. maximization problem on a convex compact. We prove that
the standard optimality condition (the complete linearization of the objective function) is equivalent to
a condition with the partial linearization with respect to the first of participating functions. We propose
two procedures for finding extremal points of the problem.
The extension of a local condition to all points of the global maximum gives a sufficient optimality
condition; one can prove it by the reduction to an equivalent convex optimization problem. As distinct
from the result obtained in [3], the established condition contains no parameter and is formulated in the
form of an inequality for the directional derivative of the objective function at feasible points on a level
surface.
2. NECESSARY OPTIMALITY CONDITION
Let D ⊂
Rn
be a convex compact set, int D = ∅. We introduce a d. c.-function
φ(x) = φ1 (x) − φ2 (x),
x ∈ D,
with quadratic strongly convex participating functions
φi (x) = 1/2x − ai , Ci (x − ai ),
ai ∈ Rn ,
Ci ∈ Rn×n , i = 1, 2,
where Ci is a symmetric positive definite matrix (Ci > 0). Denote C = C1 − C2 . This is the matrix of
second partial derivatives of the function φ(x).
*
E-mail: [email protected].
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