# Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization

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## Abstract

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.

## Mathematics Subject Classification

90C20 90C22 90C26 49N30## References

- 1.Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. Ser. B
**95**, 351–370 (2003)MathSciNetCrossRefGoogle Scholar - 2.Burer, S., Anstreicher, K.M.: Second-order cone constraints for extended trust-region problems. Preprint, Optimization online, March (2011)Google Scholar
- 3.Beck, A., Eldar, Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim.
**17**, 844–860 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Beck, A.: Convexity properties associated with nonconvex quadratic matrix functions and applications to quadratic programming. J. Optim. Theory Appl.
**142**, 1–29 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications. SIAM-MPS, Philadelphia (2000)Google Scholar
- 6.Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.:
*Robust Optimization*. Princeton Series in Applied Mathematics (2009)Google Scholar - 7.Ben-Tal, A., Nemirovski, A., Roos, C.: Robust solutions of uncertain quadratic and conic quadratic problems. SIAM J. Optim.
**13**(2), 535–560 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Bertsimas, D., Brown, D., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev.
**53**, 464–501 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Bertsimas, D., Pachamanova, D., Sim, M.: Robust linear optimization under general norms. Oper. Res. Lett.
**32**, 510–516 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS-SIAM Series in Optimization, SIAM, Philadelphia, PA (2000)Google Scholar
- 11.Dines, L.L.: On the mapping of quadratic forms. Bull. Am. Math. Soc.
**47**, 494–498 (1941)MathSciNetCrossRefGoogle Scholar - 12.Fradkov, A.L., Yakubovich, V.A.: The S-procedure and duality relations in nonconvex problems of quadratic programming. Leningrad, Russia. Vestn. Leningr. Univ.
**6**, 101–109 (1979)zbMATHGoogle Scholar - 13.El Ghaoui, L., Lebret, H.: Robust solution to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl.
**18**(4), 1035–1064 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Jeyakumar, V., Lee, G.M., Li, G.Y.: Alternative theorems for quadratic inequality systems and global quadratic optimization. SIAM J. Optim.
**20**(2), 983–1001 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Jeyakumar, V., Li, G.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim.
**20**, 3384–3407 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Jeyakumar, V., Li, G.: Exact SDP Relaxations for classes of nonlinear semidefinite programming problems. Oper. Res. Lett. (2012). doi: 10.1016/j.orl.2012.09.006
- 17.Jeyakumar, V., Huy, N.Q., Li, G.: Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization. Optim. Eng.
**10**, 491–503 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Jeyakumar, V., Li, G.: A robust von-Neumann minimax theorem for zero-sum games under bounded payoff uncertainty. Oper. Res. Lett.
**39**(2), 109–114 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Jeyakumar, V., Rubinov, A.M., Wu, Z.Y.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program. Ser. A
**110**, 521–541 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Jeyakumar, V., Wolkowicz, H.: Zero duality gaps in infinite-dimensional programming. J. Optim. Theory Appl.
**67**, 87–108 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 21.More, J.J.: Generalizations of the trust region subproblem. Optim. Methods Softw.
**2**, 189–209 (1993)CrossRefGoogle Scholar - 22.Pardalos, P., Romeijn, H.: Handbook in Global Optimization, vol. 2. Kluwer Academic Publishers, Dordrecht (2002)CrossRefGoogle Scholar
- 23.Peng, J.M., Yuan, Y.X.: Optimality conditions for the minimization of a quadratic with two quadratic constraints. SIAM J. Optim.
**7**, 579–594 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Pólik, I., Terlaky, T.: A survey of the S-Lemma. SIAM Rev.
**49**, 371–418 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 25.Polyak, B.T.: Convexity of quadratic transformation and its use in control and optimization. J. Optim. Theory Appl.
**99**, 563–583 (1998)MathSciNetCrossRefGoogle Scholar - 26.Powell, M.J.D., Yuan, Y.: A trust region algorithm for equality constrained optimization. Math. Program.
**49**(91), 189–211 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Stern, R.J., Wolkowicz, H.: Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim.
**5**, 286–313 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Sturm, J.F., Zhang, S.Z.: On cones of nonnegative quadratic functions. Math. Oper. Res.
**28**, 246–267 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 29.Yakubovich, Y.A.: The S-procedure in nonlinear control theory. Vestn. Leningr. Univ.
**4**(1), 62–77 (1971)Google Scholar - 30.Ye, Y.Y., Zhang, S.Z.: New results of quadratic minimization. SIAM J. Optim.
**14**, 245–267 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Yuan, Y.X.: On a subproblem of trust region algorithms for constrained optimization. Math. Program.
**47**, 53–63 (1990)CrossRefzbMATHGoogle Scholar