Optimization and convexity

Optimization basically comprises two worlds, whose problems are similar from a distance, but whose methods are very different: the continuous and the discrete. This issue deals mainly with non-differentiable optimization, which straddles the line between the two worlds: 100% of the methods used belong to the continuous world, but 90% of the problems are in some way related to discrete optimization.

Examples include: industrial cutting, vehicle or crew routing, multiflot routing in telecommunications, etc. Some of the most effective techniques for tackling these problems (column generation, Branch and Price) involve the kind of optimization we're talking about here: continuous and non-differentiable.

Large-scale problems belong to the same family: because of their number of variables or constraints, or because they comprise several heterogeneous elements, these problems require a special technology: decomposition, which generally leads to non-differentiable optimization. In production engineering, for example, there may be a large number of different types of means of production, all contributing to the same output: this is the case with electrical energy, produced by nuclear power plants, conventional thermal power plants and hydroelectric turbines; these means of production are very different from one another.

The main types of problem mentioned above come from the "social" sciences; others of a similar nature can be found in automatic control (stabilization), statistics (calibration of covariance matrices), mechanics (impact problems), electronics (semiconductors) – non-exhaustive list.