piecewise linear quadratic

Piecewise linear-quadratic (PLQ) functions are a class of continuous functions that are characterized by their domain being composed of a finite union of polyhedral sets. Within each of these sets, the function behaves quadratically. This means that on each polyhedral subset, the function can be represented by a quadratic polynomial, but the overall function is defined piecewise across different subsets. The concept of PLQ functions is significant in optimization problems, where they can be used to model various constraints and objectives.

PLQ functions are particularly useful in convex analysis, as the set of PLQ functions is closed under certain operations, such as addition, multiplication by a scalar, and pointwise maximum with another convex function. This property makes them a versatile tool for formulating and solving optimization problems that involve convex constraints or objectives6.

In the context of optimization, a piecewise linear-quadratic program is an optimization problem that involves a PLQ objective function or constraints. These programs can be challenging to solve due to their non-smooth nature, but recent research has focused on developing efficient algorithms for such problems. For instance, the paper by Ying Cui, Tsung Hui Chang, Mingyi Hong, and Jong Shi Pang from 2020 discusses the properties of PLQ functions and their role in optimization13.

Moreover, the study of local properties of PLQ functions, such as first and second order directional derivatives, is crucial for understanding their behavior and for developing optimization algorithms that can exploit these properties to achieve better convergence rates5. In some cases, specific conditions related to PLQ functions have been shown to be sufficient to justify the linear convergence of the primal-dual sequence generated by the augmented Lagrangian method, which is an important result for the development of optimization algorithms4.

In summary, piecewise linear-quadratic functions are an important concept in the field of optimization, particularly for problems involving convex constraints or objectives. They are defined by their continuity and piecewise quadratic behavior on polyhedral sets, and recent research has focused on understanding their properties and developing efficient algorithms for solving optimization problems involving PLQ functions.