## Invited Session Mon.3.H 2038

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

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

### Matrix optimization

**Chair: Defeng Sun**

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

**Houduo Qi**

Computing the nearest Euclidean distance matrix

**Abstract:**

The Nearest Euclidean distance matrix problem (NEDM) is a fundamental computational problem in applications such as multidimensional scaling and molecular conformation from nuclear magnetic resonance data in computational chemistry. Especially in the latter application, the problem is often a large scale with the number of atoms ranging from a few hundreds to a few thousands. In this paper, we introduce a semismooth Newton method that solves the dual problem of (NEDM). We prove that the method is quadratically convergent. We then present an application of the Newton method to NEDM with *H*-weights via majorization and an accelerated proximal gradient scheme. We demonstrate the superior performance of the Newton method over existing methods including the latest quadratic semi-definite programming solver. This research also opens a new avenue towards efficient solution methods for the molecular embedding problem.

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

**Bin Wu**

The Moreau-Yosida regularization of the Ky Fan *k*-norm related functions

**Coauthors: Chao Ding, Defeng Sun, Kim-Chuan Toh**

**Abstract:**

Matrix optimization problems (MOPs) involving the Ky Fan *k*-norm arise frequently in diverse fields such as matrix norm approximation, graph theory, and so on. In order to apply the proximal point algorithms to solve large scale MOPs involving the Ky Fan *k*-norm, we need to understand the first and second order properties of the Moreau-Yosida regularization of the Ky Fan *k*-norm function and the indicator function of its epigraph. As an initial step, we first study the counterparts of the vector *k*-norm related functions, including the metric projectors over the dual vector *k*-norm ball and the vector *k*-norm epigraph, and their directional derivatives and Fréchet differentiability. We then use these results to study the corresponding properties for the Moreau-Yosida regularization of the Ky Fan *k*-norm epigraph indicator function.

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

**Renato D. Monteiro**

An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods

**Coauthor: Benar F. Svaiter**

**Abstract:**

We present an accelerated variant of the hybrid proximal

extragradient (HPE) method for convex optimization,

referred to as the A-HPE method.

Iteration-complexity results are established for

the A-HPE method, as well as a special version of it,

where a large stepsize condition is imposed.

Two specific implementations of the A-HPE method are described in the context

of a structured convex optimization problem whose objective function consists

of the sum of a smooth convex function and an extended real-valued non-smooth

convex function. In the first implementation, a generalization of a variant of

Nesterov's method is obtained for the case where the smooth component

of the objective function has Lipschitz continuous gradient.

In the second one, an accelerated Newton proximal extragradient

(A-NPE) method is obtained for the case where the smooth component of

the objective function has Lipschitz continuous Hessian. It is shown that

the A-NPE method has a *O(1/k*^{7/2}) convergence rate,

which improves upon the *O(1/k*^{3})

convergence rate bound for another accelerated Newton-type

method presented by Nesterov.