Global Optimization Algorithm for Mixed-Integer Nonlinear Programs with Trigonometric Functions

TL;DR

The paper tackles global optimization of MINLPs that include bounded, differentiable periodic terms, with a focus on trigonometric functions. It introduces a MILP-based iterative algorithm that builds piecewise polyhedral relaxations via adaptive domain partitions and incremental formulations for trig and bilinear terms, and it leverages principal-domain reformulations to exploit periodicity. Key contributions include (i) polyhedral relaxations for univariate trig terms and bilinear terms, (ii) partition-sharing and multiple refinement schemes/strategies to accelerate convergence, and (iii) a motivating MDPPP application with extensive computational results showing improvements over original formulations and some commercial/open-source solvers. The approach offers a scalable pathway to solving a broader class of MINLPs by integrating trig/bilinear relaxations into MILP-based global optimization, potentially enabling faster and more reliable global solutions in engineering problems with periodic nonlinearities.

Abstract

This article presents the first mixed-integer linear programming (MILP)-based iterative algorithm to solve factorable mixed-integer nonlinear programs (MINLPs) with bounded, differentiable periodic functions to global optimality with an emphasis on trigonometric functions. At each iteration, the algorithm solves a MILP relaxation of the original MINLP to obtain a bound on the optimal objective value. The relaxations are constructed using partitions of variables involved in each nonlinear term and across successive iterations, the solution of the relaxations is used to refine these partitions further leading to tighter relaxations. Also, at each iteration, a heuristic/local solve on the MINLP is used to obtain a feasible solution to the MINLP. The iterative algorithm terminates till the optimality gap is sufficiently small. This article proposes novel refinement strategies that first choose a subset of variables whose domain is refined, refinement schemes that specify the manner in which the variable domains are refined, and MILP relaxations that exploit the principal domain of the periodic functions. We also show how solving the resulting MILP relaxation may be accelerated when two or more periodic functions are related by a linking constraint. This is especially useful as any periodic function may be approximated to arbitrary precision by a Fourier series. Finally, we examine the effectiveness of the proposed approach by solving a path planning problem for a single fixed-wing aerial vehicle and present extensive numerical results comparing the various refinement schemes and techniques.

Figures (15)

• Figure 1: Simplified flowchart for the proposed algorithm.
• Figure 2: Example of triangles formed by overestimates, underestimates, and secant lines of $f(x) = \sin x$ with $x \in [0, 2\pi]$ and admissible partition $p$. The points on the curve corresponding to the partition points are shown with black markers. The green triangles correspond to the points $v_i$, $v_{i, i + 1}$, and $v_{i + 1}$ for each sub-interval $[x_i, x_{i + 1}]$ of $p$. (a) Triangles formed when using base partition $p^0 = (0, \pi, 2\pi)$. (b) Triangles formed when using the admissible partition $p = (0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi)$, which is a valid refinement of the base partition $p^0$.
• Figure 3: Convex hull of $z = xy$ with domain $[-1, 1] \times [-1, 1]$.
• Figure 4: Disjunctive union of three tetrahedrons containing $z = xy$ over the domain $[-1, 1] \times [-1, 1]$. The variable $x$ has been partitioned using $p_x = (-1, -0.25, 0.25, 1)$, shown by the colored bars along the $x$-axis with tetrahedrons being colored accordingly.
• Figure 5: Example of using a shared partition, $p^0 = (0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi)$, for $f_1(x) = \sin x$ and $f_2(x) = \cos x$ over the closed interval $[0, 2\pi]$. It can be seen the $i$-th triangle for $f_1$ and the $i$-th triangle for $f_2$ are defined over the same sub-interval, so they can be linked by sharing the same binary variables in the MILP relaxations. The binary variables then indicate which sub-interval the solution is in, rather than which triangle.

Theorems & Definitions (11)

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