Semidefinite Representation of Convex Sets

Let $S =\{x\in \re^n:\, g_1(x)\geq 0, \cdots, g_m(x)\geq 0\}$ be a semialgebraic set defined by multivariate polynomials $g_i(x)$. Assume $S$ is convex, compact and has nonempty interior. Let $S_i =\{x\in \re^n:\, g_i(x)\geq 0\}$, and $\bdS$ (resp. $\bdS_i$) be the boundary of $S$ (resp. $S_i$). This paper, as does the subject of semidefinite programming (SDP), concerns … Read more

Polynomial time algorithms to approximate mixed volumes within a simply exponential factor

We study in this paper randomized algorithms to approximate the mixed volume of well-presented convex compact sets. Our main result is a randomized poly-time algorithm which approximates $V(K_1,…,K_n)$ with multiplicative error $e^n$ and with better rates if the affine dimensions of most of the sets $K_i$ are small.\\ Even such rate is impossible to achieve … Read more

Finding the projection of a point onto the intersection of convex sets via projections onto halfspaces

We present a modification of Dykstra’s algorithm which allows us to avoid projections onto general convex sets. Instead, we calculate projections onto either a halfspace or onto the intersection of two halfspaces. Convergence of the algorithm is established and special choices of the halfspaces are proposed. The option to project onto halfspaces instead of general … Read more