# Semidefinite Representation of Convex Sets - Mathematics > Optimization and Control

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Abstract: Let $S =\{x\in e^n: g 1(x)\geq 0,

., g m(x)\geq 0\}$ be a semialgebraicset defined by multivariate polynomials $g i(x)$. Assume $S$ is convex, compactand has nonempty interior. Let $S i =\{x\in e^n: g i(x)\geq 0\}$, and $\bdS$(resp. $\bdS i$) be the boundary of $S$ (resp. $S i$). This paper discusseswhether $S$ can be represented as the projection of some LMI representable set.Such $S$ is called semidefinite representable or SDP representable. Thecontributions of this paper: {\bf (i)} Assume $g i(x)$ are all concave on $S$.If the positive definite Lagrange Hessian (PDLH) condition holds, i.e., theHessian of the Lagrange function for optimization problem of minimizing anynonzero linear function $\ell^Tx$ on $S$ is positive definite at the minimizer,then $S$ is SDP representable. {\bf (ii)} If each $g i(x)$ is eithersos-concave ($- abla^2g i(x)=W(x)^TW(x)$ for some possibly nonsquare matrixpolynomial $W(x)$) or strictly quasi-concave on $S$, then $S$ is SDPrepresentable. {\bf (iii)} If each $S i$ is either sos-convex or poscurv-convex($S i$ is compact convex, whose boundary has positive curvature and isnonsingular, i.e. $ abla g i(x) ot = 0$ on $\bdS i \cap S$), then $S$ is SDPrepresentable. This also holds for $S i$ for which $\bdS i \cap S$ extendssmoothly to the boundary of a poscurv-convex set containing $S$. {\bf (iv)} Wegive the complexity of Schm\-{u}dgen and Putinar-s matrix Positivstellensatz,which are critical to the proofs of (i)-(iii).

Author: ** J. William Helton, Jiawang Nie**

Source: https://arxiv.org/