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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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