Since the advent of modern computational mathematics, the literature
on optimization algorithms has been ever-growing, and a wide range of
methods are now available. Usually, published algorithms are shown to
be applicable to given templates of optimization problems, and are
often accompanied by proofs characterizing their convergence rates.

There exist many variations of such templates. Indeed, a template may
describe the objective as a single function or a sum of functions with
different characteristics (say, smoothness, convexity, existence of a
computable proximal operator, etc.), accompanied by constraints that
can also be described in various ways (e.g. linear constraints,
feasible set with a computable projection operator, functional
constraints, etc.).

Hence, for a given instance of an optimization problem, it is not
always easy to identify which templates it can match, especially if
one wants to consider equivalent reformulations of the problem. This
makes the task of choosing an optimization method tedious for the
practitioner.

In this talk, we present a modeling language for mathematical
programming that aims at describing oracle-based optimization problem
formulations. Using this language, we propose a framework that
automatically checks whether a user-provided optimization problem fits
a known template. The framework, implemented as a Python package,
relies on an extensive library of optimization methods, described
using the above-mentioned language, with their associated known
convergence rate for one or several templates, with the goal of
automatically ranking applicable optimization methods according to
their worst-case theoretical guarantees. The framework also handles
some reformulation techniques, allowing more corresponding templates
to be potentially identified.

We describe how to detect symmetry in Mixed-Integer Conic Programming. We present our framework as an extension to the Mixed-Integer Linear Programming case, resulting in a relatively small formal overhead. To do so, we introduce the concept of symmetry labelings for cones and study their properties, with a focus on what labelings are useful in practice. We also report on computational experiments performing symmetry detection in the optimization software package MOSEK on a series of Mixed-Integer Conic Programming problems.

Hexaly is a new kind of global optimization solver that combines exact and heuristic methods. Heuristics like local search are known for delivering quality primal solutions to large-scale combinatorial optimization problems in short running times. However, these are difficult to apply when continuous decisions are involved, as in inventory routing, nuclear power plant outage scheduling, portfolio optimization with limited assets, and sparse regression. To address these problems, Hexaly automatically decomposes them into a pure combinatorial problem and a continuous subproblem parametrized by the combinatorial decisions. The combinatorial space is explored by applying local search techniques. Once a new feasible solution is found for integer decisions, an exact continuous algorithm is launched to determine continuous decisions optimally: simplex methods when the continuous subproblem is linear and interior point methods when it is nonlinear. Sensitivity analysis is performed to guide the heuristic search in the combinatorial space. The talk will present the overall approach and illustrate it on the problems previously mentioned.

In this talk we present the most recent improvements to the SAS linear and mixed-integer optimization solvers and discuss future developments. Topics include simplex implementational aspects, the latest ideas for crossover, new presolve techniques and improvement for existing techniques, and the latest developments in cut generation, selection and management.