## Invited Session Thu.3.H 3003A

#### Thursday, 15:15 - 16:45 h, Room: H 3003A

**Cluster 6: Derivative-free & simulation-based optimization** [...]

### Recent progress in direct search methods

**Chair: Luís Nunes Vicente and Stefan Wild**

**Thursday, 15:15 - 15:40 h, Room: H 3003A, Talk 1**

**Sébastien Le Digabel**

The mesh adaptive direct search algorithm with reduced number of directions

**Coauthors: Charles Audet, Andrea Ianni, Christophe Tribes**

**Abstract:**

The Mesh Adaptive Direct Search (MADS) class of algorithms is designed for blackbox optimization where the objective function and constraints are typically computed by launching a time-consuming computer simulation. The core of each iteration of the algorithm consists of launching the simulation at a finite number of trial points. These candidates are constructed from MADS directions. The current and efficient implementation of MADS uses *2n* directions at each iteration, where *n* is the number of variables. The scope of the present work is the reduction of that number to a minimal positive spanning set of *n+1* directions. This transformation is generic and can be applied to any method that generates more than *n+1* MADS directions.

**Thursday, 15:45 - 16:10 h, Room: H 3003A, Talk 2**

**José Mario Martínez**

Inexact restoration method for derivative-free optimization with smooth constraints

**Coauthors: Luís Felipe Bueno, Ana Friedlander, Francisco N. C. Sobral**

**Abstract:**

A new method is introduced for solving constrained optimization problems in which the derivatives of the constraints are available but the derivatives of the objective function are not. The method is based on the Inexact Restoration framework, by means of which each iteration is divided in two phases. In the first phase one considers only the constraints, in order to improve feasibility. In the second phase one minimizes a suitable objective function subject to a linear approximation of the constraints. The second phase must be solved using derivative-free methods. An algorithm introduced recently by Kolda, Lewis, and Torczon for linearly constrained derivative-free optimization is employed for this purpose. Under usual assumptions, convergence to stationary points is proved. A computer implementation is described and numerical experiments are presented.

**Thursday, 16:15 - 16:40 h, Room: H 3003A, Talk 3**

**Rohollah Garmanjani**

Smoothing and worst case complexity for direct-search methods in non-smooth optimization

**Coauthor: Luís Nunes Vicente**

**Abstract:**

For smooth objective functions it has been shown that the worst case cost

of direct-search methods is of the same order as the one of steepest descent.

Motivated by the lack of such a result in the non-smooth case, we propose, analyze,

and test a

class of smoothing direct-search methods for the optimization of non-smooth functions.

Given a parameterized family of smoothing functions for the non-smooth objective

function,

this class of methods consists of applying a direct search for a fixed value of the

smoothing parameter until the step size is relatively small, after which the

smoothing parameter

is reduced and the process is repeated.

One can show that the worst case complexity (or cost) of this procedure is roughly

one order

of magnitude worse than the one for direct search or steepest descent on smooth

functions.

The class of smoothing direct-search methods is also showed to enjoy asymptotic

global convergence properties.

Numerical experience indicates that this approach leads to better values of the

objective function, apparently without an additional cost in the number of function

evaluations.