Closest univariate convex linear-quadratic function approximation with minimal number of Pieces

TL;DR

The paper tackles the problem of approximating a univariate PLQ function by the closest convex PLQ function in the $L_2$ sense, while also minimizing the number of pieces in the representation. It introduces four algorithms: two that compute the closest convex PLQ with fixed breakpoints or fixed piece count, and two that reduce the piece count by optimization over the breakpoints, including a generalization to non-convex PLQ. The methods are implemented in MATLAB (YALMIP) and solved with CPLEX, Gurobi, or BARON, and are validated via numerical experiments and an application to road-design vertical alignments, with comparisons to a globally optimal univariate spline method. The results demonstrate that minimal-piece convex PLQ approximations can significantly reduce representation size and computation time in downstream tasks, while preserving accuracy, and they provide practical guidance on solver choices and reformulation opportunities. The work also releases complete code and outlines avenues for future enhancements, such as extending to multivariate PLQ and leveraging heuristic strategies.

Abstract

We compute the closest convex piecewise linear-quadratic (PLQ) function with minimal number of pieces to a given univariate piecewise linear-quadratic function. The Euclidean norm is used to measure the distance between functions. First, we assume that the number and positions of the breakpoints of the output function are fixed, and solve a convex optimization problem. Next, we assume the number of breakpoints is fixed, but not their position, and solve a nonconvex optimization problem to determine optimal breakpoints placement. Finally, we propose an algorithm composed of a greedy search preprocessing and a dichotomic search that solves a logarithmic number of optimization problems to obtain an approximation of any PLQ function with minimal number of pieces thereby obtaining in two steps the closest convex function with minimal number of pieces. We illustrate our algorithms with multiple examples, compare our approach with a previous globally optimal univariate spline approximation algorithm, and apply our method to simplify vertical alignment curves in road design optimization. CPLEX, Gurobi, and BARON are used with the YALMIP library in MATLAB to effectively select the most efficient solver.

Figures (8)

• Figure 1: PLQ function (blue) compared to its convex envelope and closest convex function.
• Figure 2: PLQ functions admitting no closest convex function.
• Figure 3: PLQ functions admitting a closest convex function after applying lemma \ref{['lemma:bounding_box']}.
• Figure 4: Approximation error vs. number of pieces. Using the nonincreasing property of the graph, we want to compute $r$ corresponding to the red point, i.e., minimal number of pieces with approximation error lower than the error tolerance $\varepsilon$.
• Figure 5: Algorithm 1.

Theorems & Definitions (17)

• Theorem 1: BAUSCHKE-24
• Corollary 2: BAUSCHKE-24
• Lemma 3
• proof
• Remark 4
• Lemma 5
• proof
• Lemma 6
• proof
• Proposition 7