Polynomial Matrix Inequality and Semidefinite Representation
Mathematics of Operations Research
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A set is called semidefinite representable or semidefinite programming (SDP) representable if it equals the projection of a higher dimensional set which is defined by some Linear Matrix Inequality (LMI). This paper discusses the semidefinite representability conditions for convex sets of the form $${S_{\mathcal {D}}(f) =\{x\in \mathcal {D} : f(x) \geq 0 \}}$$. Here, $${\mathcal {D}=\{x\in \mathbb {R}^n : g_1(x) \geq 0, \ldots, g_m(x) \geq 0 \}}$$ is a convex domain defined by some “nice” concave polynomials g i (x) (they satisfy certain concavity certificates), and f(x) is a polynomial or rational function. When f(x) is concave over $${\mathcal {D}}$$, we prove that $${S_{\mathcal {D}}(f) }$$ has some explicit semidefinite representations under certain conditions called preordering concavity or q-module concavity, which are based on the Positivstellensatz certificates for the first order concavity criteria:$$f(u) + \nabla f(u)^T(x-u) -f(x) \geq 0, \quad \forall \, x, u \in \mathcal {D}.$$ When f(x) is a polynomial or rational function having singularities on the boundary of $${S_{\mathcal {D}}(f)}$$, a perspective transformation is introduced to find some explicit semidefinite representations for $${S_{\mathcal {D}}(f)}$$ under certain conditions. In the special case n = 2, if the Laurent expansion of f(x) around one singular point has only two consecutive homogeneous parts, we show that $${S_{\mathcal {D}}(f)}$$ always admits an explicitly constructible semidefinite representation.