A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems
L is a constant matrix of rank k < n and
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 .
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.
KeywordsInterior Point Global Convergence Interior Point Method Central Path Positive Semidefinite Matrice
Unable to display preview. Download preview PDF.
- N.S. Bahvalow, Methodes Numeriques, ed. Nauka (Mir) Moscow, 1980.Google Scholar
- P.D. Domich et al., Optimal 3-dimensional methods for linear programming, National Institute of Standards and Technology, Gaithersburg, NISTIR 89–4225.Google Scholar
- M.C. Ferris, A.B. Philpott, An interior point method for semi-infinite programming, Mathematical Programming (43) 1989 pp. 257–276.Google Scholar
- F. Jarre, Interior Point Methods for convex programming, to appear in Applied Math. and Optimization, 1993.Google Scholar
- J.C. Lagarias, M.J. Todd, eds., Mathematical Developments Arising from Linear Programming, Contemporary Mathematics, Vol. 114, Amer. Math. Soc., Providence, 1989.Google Scholar
- G. Lopes, Conditions for convergence of multipoint Padé approximations for functions of Stieltjes type, Math. USSR Sbornik, Vol. 35 (1979), No. 3.Google Scholar
- I.J. Lustig, R.E. Marsten, D.F.Shanno, Computational Experience with a Primal-Dual Interior Point Method for Linear Programming Techniques, Industrial and Syst. Engineering Rep. Ser., Rep. J-89-11, Inst. of Technology, Atlanta, Georgia.Google Scholar
- U. Schättier, An interior point method for semi-infinite programming problems, Doct. Dissertation, Univ. Würzburg, Inst. f. Angew. Math. 1992.Google Scholar
- G. Sonnevend, J. Stoer, G. Zhao, On the complexity of following the central path by linear extrapolation in linear programs, Mathematical Programming, 1991, pp. 527–553.Google Scholar
- G. Sonnevend, Constructing feedback control in differential games by the use of central trajectories, DFG Report, Nr. 385/1992, Inst. für Angewandte Mathematik, Univ. Würzburg (July 1992), 37 p., to appear in ZAMM.Google Scholar
- G. Sonnevend, Applications of Analytic Centers, NATO ASI Ser. F, Vol. 70, “Numerical Linear Algebra and Digital Signal Processing” (P. van Dooren, and G.Golub eds.), Reidel 1988.Google Scholar
- G. Sonnevend, Application of analytic centers for the numerical solution of semi-infinite, convex programs arising in control theory, 15 p., DFG Report , Anwendungsbezogene Optimierung und Steuerung, Nr. 170/1989, Inst. für Angewandte Mathematik, Univ,. Würzburg.Google Scholar
- T. Tsuchiya, M. Murarmatsu, Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems, Res. Memo. Nr. 423, Inst. of Statistical Math., Tokyo 1992.Google Scholar