## Invited Session Thu.3.H 2038

#### Thursday, 15:15 - 16:45 h, Room: H 2038

**Cluster 4: Conic programming** [...]

### Recent developments of theory and applications in conic optimization I

**Chair: Hayato Waki and Masakazu Muramatsu**

**Thursday, 15:15 - 15:40 h, Room: H 2038, Talk 1**

**Muddappa Seetharama Gowda**

On the nonhomogeneity and the bilinearity rank of a completely positive cone

**Abstract:**

Given a closed cone *C* in *R*^{n}, the completely positive

cone of *C* is the convex cone *K*

generated by matrices of the form *uu*^{T} as *u* varies over *C*. Examples of completely positive cones include the

positive semidefinite cone (when *C=R*^{n}) and

the cone of completely positive matrices (when *C=R*^{n}_{+}). Completely positive cones arise,

for example, in the reformulation of a nonconvex quadratic minimization problem over an arbitrary set with linear and binary constraints as

a conic linear program. This talk deals with the questions of

when (or whether) *K* is self-dual, irreducible, and/or homogeneous. We also describe

the blinearity rank of *K* in terms of that of *C*.

**Thursday, 15:45 - 16:10 h, Room: H 2038, Talk 2**

**Masakazu Muramatsu**

A perturbed sums of squares theorem for polynomial optimization and its applications

**Coauthors: Levent Tuncel, Hayato Waki**

**Abstract:**

We prove a property of positive polynomials on a compact set with a small perturbation. When applied to a POP, the property implies that the optimal value of the corresponding SDP relaxation with sufficiently large order is bounded below by *f*^{\}ast-ε and from above by *f*^{\}ast + ε(n+1), where *f*^{\}ast is the optimal value of the POP, *n* is the number of variables, and *ε* is the perturbation. In addition to extending this property to some directions, we propose a new sparse SDP relaxation based on it. In this relaxation, we positively exploit the numerical errors naturally introduced by numerical computation. An advantage of our SDP relaxation is that they are of considerably smaller dimensional than Lasserre's, and in many situation than the sparse SDP relaxation proposed by Waki et al. We present some applications and the results of our computational experiments.

**Thursday, 16:15 - 16:40 h, Room: H 2038, Talk 3**

**Farid Alizadeh**

Some geometric applications of abstract algebraic sum-of-squares cones

**Abstract:**

We have established that sum-of-squares (SOS) cones in abstract algebras are semidefinite representable. By combining this fact and the classical theorem of Youla, it can be shown that a wide variety of cones are in fact SOS with respect to some algebra, and thus SD-representable. We review some applications of such cones in geometric optimization. We examine the minimum volume ellipsoid, the minimum volume rectangular box, the minimum volume simplex, etc., containing a space curve. We also examine the diameter of a space curve, distance of a point to a space curve, and possible optimization problems with constraints on such parameters. For instance we examine design of a space curve with constraints on its curvature. We will also comment on similar problem for higher dimensional surfaces.