On Nesterov's Approach to Semi-infinite Programming

We generalize Nesterov's construction for the reduction of various classes of semi-infinite programming problems to the semidefinite programming form. In this way, we are able to consider ‘cones of squares’ of real-valued and matrix-valued functions as rather particular cases of a unifying abstract scheme. We also interpret from this viewpoint some results of M. Krein and A. Nudelman. This provides (in a way which probably has not been anticipated by these authors) a very powerful tool for solving various optimization problems.

[1]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[2]  Jacques Tits,et al.  Sur la trialité et certains groupes qui s’en déduisent , 1959 .

[3]  Jacques Tits,et al.  Les groupes simples de Suzuki et de Ree , 1960 .

[4]  M. G. Kreĭn,et al.  Some questions in the theory of moments , 1962 .

[5]  I. G. MacDonald,et al.  Lectures on Lie Groups and Lie Algebras: Simple groups of Lie type , 1995 .

[6]  M. Simonovits,et al.  Cycles of even length in graphs , 1974 .

[7]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[8]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[9]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[10]  G. A. Margulis,et al.  Explicit constructions of graphs without short cycles and low density codes , 1982, Comb..

[11]  Norman Biggs,et al.  The sextet construction for cubic graphs , 1983, Comb..

[12]  Wilfried Imrich,et al.  Explicit construction of regular graphs without small cycles , 1984, Comb..

[13]  Alfred Weiss.  Girths of bipartite sextet graphs , 1984, Comb..

[14]  N. Koblitz.  A Course in Number Theory and Cryptography , 1987 .

[15]  A. Neumaier,et al.  Distance Regular Graphs , 1989 .

[16]  Norman L. Biggs,et al.  Note on the girth of Ramanujan graphs , 1990, J. Comb. Theory, Ser. B.

[17]  Andries E. Brouwer,et al.  The complement of a geometric hyperplane in a generalized polygon is usually connected , 1993 .

[18]  Moshe Morgenstern,et al.  Existence and Explicit Constructions of q + 1 Regular Ramanujan Graphs for Every Prime Power q , 1994, J. Comb. Theory, Ser. B.

[19]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[20]  Alexander Lubotzky,et al.  Discrete groups, expanding graphs and invariant measures , 1994, Progress in mathematics.

[21]  V. A. Ustimenko,et al.  On Some Properties of Geometries of Chevalley Groups and Their Generalizations , 1994 .

[22]  Alexander Lubotzky,et al.  Cayley graphs: eigenvalues, expanders and random walks , 1995 .

[23]  F. Lazebnik,et al.  A new series of dense graphs of high girth , 1995, math/9501231.

[24]  Felix Lazebnik,et al.  Explicit Construction of Graphs with an Arbitrary Large Girth and of Large Size , 1995, Discret. Appl. Math..

[25]  Felix Lazebnik,et al.  A characterization of the components of the graphs D(k, q) , 1996, Discret. Math..

[26]  Algebras and Lie Groups , 1997 .

[27]  Neal Koblitz,et al.  Algebraic aspects of cryptography , 1998, Algorithms and computation in mathematics.

[28]  Yurii Nesterov,et al.  Squared Functional Systems and Optimization Problems , 2000 .

[29]  S. Landau.  Standing the Test of Time : The Data Encryption Standard , 2000 .

[30]  Paul Van Dooren,et al.  Convex optimization over positive polynomials and filter design , 2000 .

[31]  Vasyl Ustimenko,et al.  CRYPTIM: Graphs as Tools for Symmetric Encryption , 2001, AAECC.

[32]  Zhi-Quan Luo,et al.  Linear matrix inequality formulation of spectral mask constraints , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[33]  Leonid Faybusovich.  Self-Concordant Barriers for Cones Generated by Chebyshev Systems , 2002, SIAM J. Optim..

[34]  Paul Van Dooren,et al.  Optimization Problems over Positive Pseudopolynomial Matrices , 2003, SIAM J. Matrix Anal. Appl..

[35]  Vasiliy A. Ustimenko,et al.  Extremal properties of regular and affine generalized m-gons as tactical configurations , 2003, Eur. J. Comb..