Generic Properties for Semialgebraic Programs

In this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints, in which every problem of the class is obtained by linear perturbations of the objective function, while the “core” objective function and the constraint functions are kept fixed. Assume that the linear independence constraint qualification is satisfied at every point in the constraint set. It is shown that almost all problems in the class are such that (i) the restriction of the objective function on the constraint set is coercive and regular at infinity; (ii) there is a unique optimal solution, lying on a unique active manifold, at which the strict complementarity and second order sufficiency conditions, the quadratic growth condition, and the Holder type global error bound hold, and (iii) all minimizing sequences converge. Furthermore, the active manifold is constant, and the optimal solution and the optimal value function depend analytically under local perturbations of the ...

[1]  Michael L. Overton,et al.  Complementarity and nondegeneracy in semidefinite programming , 1997, Math. Program..

[2]  Ha Huy Vui,et al.  Representations of Positive Polynomials and Optimization on Noncompact Semialgebraic Sets , 2010, SIAM J. Optim..

[3]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[4]  Adrian S. Lewis,et al.  Generic Optimality Conditions for Semialgebraic Convex Programs , 2011, Math. Oper. Res..

[5]  N. Z. Shor.  An approach to obtaining global extremums in polynomial mathematical programming problems , 1987 .

[6]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[7]  Okitsugu Fujiwara.  A Note on Differentiability of Global Optimal Values , 1985, Math. Oper. Res..

[8]  L. Dries,et al.  Geometric categories and o-minimal structures , 1996 .

[9]  James Demmel,et al.  Minimizing Polynomials via Sum of Squares over the Gradient Ideal , 2004, Math. Program..

[10]  Pablo A. Parrilo,et al.  Minimizing Polynomial Functions , 2001, Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science.

[11]  Huy Vui Hà,et al.  Solving polynomial optimization problems via the truncated tangency variety and sums of squares , 2009 .

[12]  Gábor Pataki,et al.  On the generic properties of convex optimization problems in conic form , 2001, Math. Program..

[13]  Okitsugu Fujiwara.  Morse Programs: A Topological Approach to Smooth Constrained Optimization , 1982, Math. Oper. Res..

[14]  Raul E. Curto,et al.  Truncated K-moment problems in several variables , 2005 .

[15]  Sean A Broughton,et al.  Milnor numbers and the topology of polynomial hypersurfaces , 1988 .

[16]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[17]  I. Ekeland.  Nonconvex minimization problems , 1979 .

[18]  Ha Huy Vui,et al.  Global Optimization of Polynomials Using the Truncated Tangency Variety and Sums of Squares , 2008, SIAM J. Optim..

[19]  A. D. Ioffe,et al.  An Invitation to Tame Optimization , 2008, SIAM J. Optim..

[20]  J. Risler,et al.  Real algebraic and semi-algebraic sets , 1990 .

[21]  Gue Myung Lee,et al.  Stability and Genericity for Semi-algebraic Compact Programs , 2016, J. Optim. Theory Appl..

[22]  Alexander D. Ioffe,et al.  Almost Every Convex or Quadratic Programming Problem Is Well Posed , 2004, Math. Oper. Res..

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

[24]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[25]  Panos M. Pardalos,et al.  Quadratic programming with one negative eigenvalue is NP-hard , 1991, J. Glob. Optim..

[26]  Jiawang Nie,et al.  Certifying convergence of Lasserre’s hierarchy via flat truncation , 2011, Math. Program..

[27]  Romesh Saigal,et al.  Generic properties of the complementarity problem , 1973, Math. Program..

[28]  A. Ivic.  Sums of squares , 2020, An Introduction to 𝑞-analysis.

[29]  Jiawang Nie,et al.  An exact Jacobian SDP relaxation for polynomial optimization , 2010, Math. Program..

[30]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[31]  Jiawang Nie,et al.  Optimality conditions and finite convergence of Lasserre’s hierarchy , 2012, Math. Program..

[32]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[33]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[34]  T. Zolezzi,et al.  Well-Posed Optimization Problems , 1993 .

[35]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[36]  Dang Tuan Hiep.  Representations of non-negative polynomials via KKT ideals , 2011 .

[37]  Markus Schweighofer,et al.  Optimization of Polynomials on Compact Semialgebraic Sets , 2005, SIAM J. Optim..

[38]  K. Schmüdgen.  TheK-moment problem for compact semi-algebraic sets , 1991 .

[39]  Markus Schweighofer.  Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares , 2006, SIAM J. Optim..

[40]  Hubertus Th. Jongen,et al.  Nonlinear Optimization in Finite Dimensions - Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects , 2000 .

[41]  Monique Laurent,et al.  Semidefinite Approximations for Global Unconstrained Polynomial Optimization , 2005, SIAM J. Optim..

[42]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[43]  M. Marshall,et al.  Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization , 2009, Canadian Journal of Mathematics.

[44]  V. Powers,et al.  Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals , 2007 .

[45]  R. T. Rockafellar,et al.  The Generic Nature of Optimality Conditions in Nonlinear Programming , 1979, Math. Oper. Res..

[46]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[47]  Jonathan E. Spingarn.  On optimality conditions for structured families of nonlinear programming problems , 1982, Math. Program..

[48]  M. Laurent.  Sums of Squares, Moment Matrices and Optimization Over Polynomials , 2009 .

[49]  Dmitriy Drusvyatskiy,et al.  Generic Minimizing Behavior in Semialgebraic Optimization , 2015, SIAM J. Optim..

[50]  M. Marshall.  Positive polynomials and sums of squares , 2008 .

[51]  Alexander Shapiro,et al.  First and second order analysis of nonlinear semidefinite programs , 1997, Math. Program..

[52]  Tien Son Pham,et al.  A Frank–Wolfe type theorem for nondegenerate polynomial programs , 2014, Math. Program..