A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems

Abstract

We shall deal with the numerical solution of optimization problems of the following type

$$ \matrix{ {\min \{ {f_0}(y){f_i}({x_i},y) \ge 0,} & {{x_i} \in [{a_i},{b_i}],} & {i = 1, \ldots ,m,} & {y \in {R^n}\} } \cr } $$

((1.1))

$$ \matrix{ {Ly = d,} & {L \in {R^{k \times n}}} \cr } $$

((1.2))

where

• 1. L is a constant matrix of rank k < n and

• 2. the semi-infinite constraint functions ƒ ₁,. . . , ƒ _m are assumed to be concave in y (at least on the domain where they are positive, see the function (det(y))^1/n on y ≥ 0, i.e. on the set of positive semidefinite matrices). Note that “finite” constraints ƒ _j (x) ≥ 0 are formally incorporated by just selecting a _j = b _j .

• 3. The functions ƒ _i , i = 0, . . .,m are assumed to be algebraically simple and rather smooth, e.g. analytic on their domains of definition, both in x and y. This condition may seem rather stringent at first, it is however of great importance. Analyticity is, of course, a very broad property. We shall show that for important classes of constraints the arising analytic functions are of special type, e. g. like the Stieltjes functions, such that they can be approximated very effectively by suitable, low complexity algorithms.