Further Results on Approximating Nonconvex Quadratic Optimization by Semidefinite Programming Relaxation

Abstract

We study approximation bounds for the semidefinite programming (SDP) relaxation ofquadratically constrained quadratic optimization: $\min f^0(x)$ subject to $f^k(x)\le 0$, $k=1,\dots,m$, where fk(x)=xTAkx+(bk)Tx+ck. In the special case of ellipsoid constraints with interior feasible solution at 0, we show that the SDP relaxation, coupled with a rank-1 decomposition result of Sturm and Zhang [Math. Oper. Res., to appear], yields a feasible solution of the original problem with objective value at most $(1-\gamma)^2/(\sqrt{m}+\gamma)^2$ times the optimal objective value, where $\gamma=\sqrt{\smash[b]{\max_k f^k(0)+1}}$. For the single trust-region problem corresponding to m=1, this yields an exact optimal solution. In the general case, we extend some bounds derived by Nesterov [Optim. Methods Softw., 9 (1998), pp. 141--160; working paper, CORE, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 1998], Ye [Math. Program., 84 (1999), pp. 219--226], and Nesterov, Wolkowicz, and Ye [in Handbook of Semidefinite Programming, H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, pp. 360--419] for the special case where $A^k$ is diagonal and bk =0 for k=1, ..., m. We also discuss the generation of approximate solutions with high probability.