Qualification Conditions in Semialgebraic Programming
Abstract
For an arbitrary finite family of semialgebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbations of this set. In particular we prove that all positive diagonal perturbations, save perhaps a finite number of them, ensure that any point within the feasible set satisfies the Mangasarian--Fromovitz constraint qualification. Using the Milnor--Thom theorem, we provide a bound for the number of singular perturbations when the constraints are polynomial functions. Examples show that the order of magnitude of our exponential bound is relevant. Our perturbation approach provides a simple protocol to build sequences of “regular” problems approximating an arbitrary semialgebraic/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
M. Abril Bucero and B. Mourrain, Exact Relaxation for Polynomial Optimization on Semi-Algebraic Sets, preprint, arXiv.1307.6426, 2014.
2.
E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer Ser. Comput. Math. 13, Springer, Berlin, 1990, https://doi.org/10.1007/978-3-642-61257-2.
3.
A. Auslender, An extended sequential quadratically constrained quadratic programming algorithm for nonlinear, semidefinite, and second-order cone programming, J. Optim. Theory Appl., 156 (2013), pp. 183--212, https://doi.org/10.1007/s10957-012-0145-z.
4.
R. Benedetti, F. Loeser, and J.-J. Risler, Bounding the number of connected components of a real algebraic set, Discrete Comput. Geom., 6 (1991), pp. 191--209, https://doi.org/10.1007/BF02574685.
5.
R. Benedetti and J.-J. Risler, Real Algebraic and Semi-Algebraic Sets, Actualités Math., Hermann, Paris, 1990.
6.
D. P. Bertsekas, Nonlinear Programming, Athena Sci. Optim. Comput. Ser., 3rd ed., Athena Scientific, Belmont, MA, 2016.
7.
L. Blum, M. Shub, and S. Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc. (N.S.), 21 (1989), pp. 1--46, https://doi.org/10.1090/S0273-0979-1989-15750-9.
8.
J. Bolte, A. Daniilidis, and A. Lewis, The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., 17 (2007), pp. 1205--1223, https://doi.org/10.1137/050644641.
9.
J. Bolte, A. Daniilidis, and A. Lewis, Tame functions are semismooth, Math. Program., 117 (2009), pp. 5--19, https://doi.org/10.1007/s10107-007-0166-9.
10.
J. Bolte, A. Daniilidis, A. Lewis, and M. Shiota, Clarke subgradients of stratifiable functions, SIAM J. Optim., 18 (2007), pp. 556--572, https://doi.org/10.1137/060670080.
11.
J. Bolte, A. Daniilidis, and A. S. Lewis, Generic optimality conditions for semialgebraic convex programs, Math. Oper. Res., 36 (2011), pp. 55--70, https://doi.org/10.1287/moor.1110.0481.
12.
J. Bolte and E. Pauwels, Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs, Math. Oper. Res., 41 (2016), pp. 442--465, https://doi.org/10.1287/moor.2015.0735.
13.
J.-F. Bonnans, J. C. Gilbert, C. Lemaréchal, and C. A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, Universitext, 2nd ed., Springer, Berlin, 2006.
14.
J.-F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Ser. Oper. Res., Springer, New York, 2000.
15.
J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization, CMS Books Math./Ouvrages de Math. SMC 3, 2nd ed., Springer, New York, 2006, https://doi.org/10.1007/978-0-387-31256-9.
16.
M. Coste, An Introduction to O-Minimal Geometry, RAAG notes, Institut de Recherche Mathématique de Rennes, 1999; available at https://perso.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf.
17.
D. A. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergrad. Texts Math., 4th ed., Springer, New York, 2015, https://doi.org/10.1007/978-3-319-16721-3.
18.
A. Daniilidis and J. C. H. Pang, Continuity and differentiability of set-valued maps revisited in the light of tame geometry, J. Lond. Math. Soc., 83 (2011), pp. 637--658.
19.
J. Demmel, J. Nie, and V. Powers, Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals, J. Pure Appl. Algebra, 209 (2007), pp. 189--200, https://doi.org/10.1016/j.jpaa.2006.05.028.
20.
J. Demmel, J. Nie, and B. Sturmfels, Minimizing polynomials via sum of squares over the gradient ideal, Math. Program., 106 (2006), pp. 587--606, https://doi.org/10.1007/s10107-005-0672-6.
21.
A. L. Dontchev, M. Quincampoix, and N. Zlateva, Aubin criterion for metric regularity, J. Convex Anal., 13 (2006), pp. 281--297.
22.
A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer Ser. Oper. Res. Financ. Eng., 2nd ed., Springer, New York, 2014, https://doi.org/10.1007/978-1-4939-1037-3.
23.
L. van den Dries, Tame Topology and O-Minimal Structures, London Math. Soc. Lecture Note Ser. 248, Cambridge University Press, Cambridge, 1998, https://doi.org/10.1017/CBO9780511525919.
24.
L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J., 84 (1996), pp. 497--540, https://doi.org/10.1215/S0012-7094-96-08416-1.
25.
D. Drusvyatskiy, A. D. Ioffe, and A. S. Lewis, Generic minimizing behavior in semialgebraic optimization, SIAM J. Optim., 26 (2016), pp. 513--534, https://doi.org/10.1137/15M1020770.
26.
R. Fletcher, An $\ell_1$ penalty method for nonlinear constraints, in Numerical Optimization 1984, SIAM, Philadelphia, PA, 1985, pp. 26--40.
27.
H. Frankowska, Some inverse mapping theorems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), pp. 183--234.
28.
H. Frankowska and M. Quincampoix, Hölder metric regularity of set-valued maps, Math. Program., 132 (2012), pp. 333--354, https://doi.org/10.1007/s10107-010-0401-7.
29.
O. Fujiwara, Morse programs: A topological approach to smooth constrained optimization. ŭshape I, Math. Oper. Res., 7 (1982), pp. 602--616.
30.
P. E. Gill and E. Wong, Sequential Quadratic Programming Methods, Springer, New York, 2012, pp. 147--224, https://doi.org/10.1007/978-1-4614-1927-3_6.
31.
H. V. Hà and T. S. Pham, Genericity in Polynomial Optimization, Ser. Optim. Appl. 3, World Scientific, Hackensack, NJ, 2017.
32.
A. Ioffe, A Sard theorem for tame set-valued mappings, J. Math. Anal. Appl., 335 (2007), pp. 882--901, https://doi.org/10.1016/j.jmaa.2007.01.104.
33.
J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796--817, https://doi.org/10.1137/S1052623400366802.
34.
J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imp. Coll. Press Optim. Ser. 1, Imperial College Press, London, 2010.
35.
G. M. Lee and T.-S. Pham, Generic properties for semialgebraic programs, Optim. Online, 2015; available at http://www.optimization-online.org/DB_FILE/2015/06/4957.pdf.
36.
G. M. Lee and T.-S. Pham, Stability and genericity for semi-algebraic compact programs, J. Optim. Theory Appl., 169 (2016), pp. 473--495, https://doi.org/10.1007/s10957-016-0910-5 https://doi.org/10.1007/s10957-016-0910-5.
37.
P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika, 17 (1970), pp. 179--184, https://doi.org/10.1112/S0025579300002850.
38.
J. W. Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville, VA, 1965.
39.
B. S. Mordukhovich, Maximum principle in the problem of time optimal response with nonsmooth constraints, Prikl. Mat. Meh., 40 (1976), pp. 1014--1023.
40.
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I, Grundlehren Math. Wiss. 330, Springer, Berlin, 2006.
41.
J. Nie, Optimality conditions and finite convergence of Lasserre's hierarchy, Math. Program., 146 (2014), pp. 97--121, https://doi.org/10.1007/s10107-013-0680-x.
42.
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Ser. Oper. Res. Financ. Eng., 2nd ed., Springer, New York, 2006.
43.
R. T. Rockafellar and R. J.-B. Wets, Variational analysis, Grundlehren Math. Wiss. 317, Springer, Berlin, 1998, https://doi.org/10.1007/978-3-642-02431-3.
44.
K. Schmüdgen, The $K$-moment problem for compact semi-algebraic sets, Math. Ann., 289 (1991), pp. 203--206, https://doi.org/10.1007/BF01446568.
45.
S. Scholtes and M. Stöhr, How stringent is the linear independence assumption for mathematical programs with complementarity constraints?, Math. Oper. Res., 26 (2001), pp. 851--863, https://doi.org/10.1287/moor.26.4.851.10007.
46.
R. Seidel, The upper bound theorem for polytopes: An easy proof of its asymptotic version, Comput. Geom., 5 (1995), pp. 115--116, https://doi.org/10.1016/0925-7721(95)00013-Y.
47.
J. E. Spingarn and R. T. Rockafellar, The generic nature of optimality conditions in nonlinear programming, Math. Oper. Res., 4 (1979), pp. 425--430, https://doi.org/10.1287/moor.4.4.425.
Information & Authors
Information
Published In
Copyright
© 2018, Society for Industrial and Applied Mathematics.
History
Submitted: 9 June 2017
Accepted: 26 February 2018
Published online: 28 June 2018
Keywords
MSC codes
Authors
Funding Information
PGMO
Air Force Office of Scientific Research https://doi.org/10.13039/100000181 : FA9550-15-1-0500
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- On continuous selections of polynomial functionsOptimization, Vol. 73, No. 2 | 9 August 2022
- Existence of Pareto Solutions for Vector Polynomial Optimization Problems with ConstraintsJournal of Optimization Theory and Applications, Vol. 195, No. 1 | 22 July 2022
- Advanced Numerical Methods Based on OptimizationNumerical Methods for Energy Applications | 23 March 2021
- Spectral Operators of Matrices: Semismoothness and Characterizations of the Generalized JacobianSIAM Journal on Optimization, Vol. 30, No. 1 | 20 February 2020
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].