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S-lemma with equality and its applications. (English) Zbl 1333.90086
Summary: Let $$f(x)=x^TAx+2a^Tx+c$$ and $$h(x)=x^TBx+2b^Tx+d$$ be two quadratic functions having symmetric matrices $$A$$ and $$B$$. The S-lemma with equality asks when the unsolvability of the system $$f(x)<0$$, $$h(x)=0$$ implies the existence of a real number $$\mu$$ such that $$f(x)+\mu h(x)\geq 0$$, $$\forall x\in\mathbb R^n$$. The problem is much harder than the inequality version which asserts that, under Slater condition, $$f(x)<0$$, $$h(x)\leq 0$$ is unsolvable if and only if $$f(x)+\mu h(x)\geq 0$$, $$\forall x\in\mathbb R^n$$ for some $$\mu\geq 0$$. In this paper, we show that the S-lemma with equality does not hold only when the matrix $$A$$ has exactly one negative eigenvalue and $$h(x)$$ is a non-constant linear function $$(B=0, b\neq 0)$$. As an application, we can globally solve $$\inf\{f(x):h(x)=0\}$$ as well as the two-sided generalized trust region subproblem $$\inf\{f(x):l\leq h(x)\leq u\}$$ without any condition. Moreover, the convexity of the joint numerical range $$\{(f(x),h_1(x),\dots,h_p(x)):x\in\mathbb R^n\}$$ where $$f$$ is a (possibly non-convex) quadratic function and $$h_1(x),\dots,h_p(x)$$ are affine functions can be characterized using the newly developed S-lemma with equality.

##### MSC:
 90C20 Quadratic programming 90C22 Semidefinite programming 90C26 Nonconvex programming, global optimization
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##### References:
 [1] Anstreicher, KM; Wright, MH, A note on the augmented hessian when the reduced Hessian is semidefinite, SIAM J. Optim., 11, 243-253, (2000) · Zbl 1047.90040 [2] Beck, A, On the convexity of a class of quadratic mappings and its application to the problem of finding the smallest ball enclosing a given intersection of ball, J. Global Optim., 39, 113-126, (2007) · Zbl 1151.90036 [3] Beck, A; Eldar, YC, Strong duality in nonconvex quadratic optimization with two quadratic constraint, SIAM J. Optim., 17, 844-860, (2006) · Zbl 1128.90044 [4] Ben-Tal, A; Hertog, D, Hidden conic quadratic representation of some nonconvex quadratic optimization problems, Math. Program. Ser. A., 143, 1-9, (2014) · Zbl 1295.90036 [5] Ben-Tal, A; Teboulle, M, Hidden convexity in some nonconvex quadratically constrained quadratic programming, Math. Program., 72, 51-63, (1996) · Zbl 0851.90087 [6] Brickman, L, On the field of values of a matrix, Proc. Am. Math. Soc., 12, 61-66, (1961) · Zbl 0104.01204 [7] Derinkuyu, K; Pınar, MÇ, On the S-procedure and some variants, Math. Meth. Oper. Res., 64, 55-77, (2006) · Zbl 1115.93025 [8] Dines, LL, On the mapping of quadratic forms, Bull. Am. Math. Soc., 47, 494-498, (1941) · Zbl 0027.15004 [9] Fang, S.C., Gao, D.Y., Lin, G.X., Sheu, R.L., Xing, W.: Double well potential function and its optimization in the n-dimenstional real space—Part I. Math. Mech. Solids (2015). doi:10.1177/1081286514566704 [10] Feng, JM; Lin, GX; Sheu, RL; Xia, Y, Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint, J. Global Optim., 54, 275-293, (2012) · Zbl 1281.90032 [11] Finsler, P, Über das vorkommen definiter und semidefiniter formen in scharen quadratischer formen, Comment. Math. Helv., 9, 188-192, (1937) · Zbl 0016.19901 [12] Fradkov, AL; Yakubovich, VA, The S-procedure and the duality relation in convex quadratic programming problems, Vestnik Leningrad. Univ., 1, 81-87, (1973) · Zbl 0259.93033 [13] Hestenes, M.R.: Optimization Theory. Wiley, New York (1975) · Zbl 0327.90015 [14] Hmam, H.: Quadratic optimization with one quadratic equality constraint. Electronic Warfare and Radar Division DSTO Defence Science and Technology Organisation, Australia, Report DSTO-TR-2416 (2010) · Zbl 1128.90046 [15] Horn, R., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985) · Zbl 0576.15001 [16] Hsia, Y; Lin, GX; Sheu, RL, A revisit to quadratic programming with one inequality quadratic constraint via matrix pencil, Pac. J. Optim., 10, 461-481, (2014) · Zbl 1327.90168 [17] Jerrard, RL, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal., 30, 721-746, (1999) · Zbl 0928.35045 [18] Jeyakumar, V, Farkas lemma: generalizations, Encycl. Optim., 2, 87-91, (2000) [19] Jeyakumar, V; Huy, NQ; Li, G, Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization, Optim. Eng., 10, 491-503, (2009) · Zbl 1273.90141 [20] Jeyakumar, V; Lee, GM; Li, GY, Alternative theorems for quadratic inequality systems and global quadratic optimization, SIAM J. Optim., 20, 983-1001, (2009) · Zbl 1197.90315 [21] Jeyakumar, V; Li, GY, Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization, Math. Program., 147, 171-206, (2014) · Zbl 1297.90105 [22] Martínez-Legaz, JE, On brickman’s theorem, J. Convex Anal., 12, 139-143, (2005) · Zbl 1077.15024 [23] Moré, JJ, Generalizations of the trust region problem, Optim. Methods Softw., 2, 189-209, (1993) [24] Nguyen, VB; Sheu, RL; Xia, Y, An SDP approach for quadratic fractional problems with a two-sided quadratic constraint, Optim. Methods Softw., (2015) · Zbl 1385.90027 [25] Palanthandalam-Madapusi, HJ; Pelt, THV; Bernstein, DS, Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint, J. Global Optim., 45, 533-549, (2009) · Zbl 1201.90148 [26] Pólik, I; Terlaky, T, A survey of the S-lemma, SIAM Rev., 49, 371-418, (2007) · Zbl 1128.90046 [27] Polyak, BT, Convexity of quadratic transformations and its use in control and optimization, J. Optim. Theory App., 99, 553-583, (1998) · Zbl 0961.90074 [28] Pong, TK; Wolkowicz, H, The generalized trust region subprobelm, Comput. Optim. Appl., 58, 273-322, (2014) · Zbl 1329.90100 [29] Shor, NZ, Quadratic optimization problems, Sov. J. Comput. Syst. Sci., 25, 1-11, (1987) · Zbl 0655.90055 [30] Sturm, JF; Wolkowicz, H, Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations, SIAM J. Optim., 5, 286-313, (1995) · Zbl 0846.49017 [31] Sturm, JF; Zhang, S, On cones of nonnegtive quadratic functions, Math. Oper. Res., 28, 246-267, (2003) · Zbl 1082.90086 [32] Tuy, H; Tuan, HD, Generalized S-lemma and strong duality in nonconvex quadratic programming, J. Global Optim., 56, 1045-1072, (2013) · Zbl 1300.90020 [33] Wang, S., Xia, Y.: Strong duality for generalized trust region subproblem: S-lemma with interval bounds. Optim. Lett. (2014). doi:10.1007/s11590-014-0812-0 · Zbl 1354.90089 [34] Xia, Y., Sheu, R.L., Fang, S.C., Xing, W.: Double well potential function and its optimization in the n-dimenstional real space—Part II. Math. Mech. Solids (2015). doi:10.1177/1081286514566723 · Zbl 1128.90044 [35] Yakubovich, VA, S-procedure in nonlinear control theory, Vestnik Leningrad. Univ., 1, 62-77, (1971) · Zbl 0232.93010 [36] Yakubovich, VA, S-procedure in nonlinear control theory, Vestnik Leningrad. Univ., 4, 73-93, (1977) [37] Ye, Y; Zhang, S, New results on quadratic minimization, SIAM J. Optim., 14, 245-267, (2003) · Zbl 1043.90064
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