Interior-point algorithms for semi-infinite programming
This paper is concerned with linear semi-infinite programming problems of the following kind: Given \(a: T\to \mathbb{R}^m\), \(\gamma: T\to \mathbb{R}\), \(b\in \mathbb{R}^m\), where \(T\) is a compact measurable subset of some \(\mathbb{R}^p\) with measure 1, the linear functional \(b^T y\), \(y\in \mathbb{R}^m\), has to be maximized subject to \(a(t)^T y\leq \gamma(t)\) for all \(t\in T\). As corresponding dual problem the following is considered: Let \(a\in C(T, \mathbb{R}^m)\) and \(\gamma\in C(T)\). Then the linear functional \(\int_T \gamma(t) \xi(t)dt\), \(\xi\in C(T)\), has to be minimized subject to \[ \int_T \xi(t) a(t)dt= b,\;\xi(t)\geq 0\qquad\text{for all}\quad t\in T. \] These problems are discretized by partitioning \(T\) into \(n\) subregions \(T_j\), \(j=1,\dots, n\), of measure \({1\over n}\). In each \(T_j\) some \(t_j\in T_j\) is chosen and for \(f\in C(T)\) the integral \(\int_T f(t)dt\) is replaced by \({1\over n}\sum^n_{j= 1} f(t_j)\). Then the discretized problem consists of maximizing \(b^T y\), \(y\in \mathbb{R}^m\), subject to \(A^T y\leq c\), where \(A= (a(t_1),\dots, a(t_n))\) and \(c= (\gamma(t_1),\dots, \gamma(t_n))^T\). The discretized dual problem consists of minimizing \(c^T x\), \(x\in \mathbb{R}^n\) subject to \(Ax= b\) and \(x\geq \theta_n\) and is the dual of the discretized problem. The vectors \(x\in \mathbb{R}^n\) then correspond to \({1\over n} (\xi(t_1),\dots, \xi(t_n))^T\) with \(\xi\in C(t)\). If \(n\) is replaced by \(2n\), the discretized problem is close to the problem of maximizing \(b^T y\), \(y\in \mathbb{R}^m\) subject to \(A^T y\leq c\), \(A^Ty\leq c\). The discretized dual problem is close to the problem of minimizing \(c^T x'+ c^T x''\), \(x'\), \(x''\in \mathbb{R}^n\), subject to \(Ax'+ Ax''= b\), \(x',x''\geq \theta_n\). If \(x\) is a feasible solution of the discretized dual problem with respect to \(n\), then \((x', x'')\) with \(x_j'= x_j''= {1\over 2} x_j\) is taken as feasible solution of the discretized dual problem with respect to \(2n\). The main concern of this paper is the question which interior point algorithm for solving the discretized problems is invariant under the operation of doubling the constraints and variables of the discretized problem and its dual, respectively. Invariance means that, given corresponding starting points, the algorithm generates corresponding iterates when applied to the \(n\)- and \(2n\)- discretized problem or its dual. It is shown that primal and dual affine-scaling and projective-scaling algorithms, as well as some large-step path-following and potential- reduction methods are invariant.
scientific article; zbMATH DE number 644756
scientific article