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A Comparative Study of MINLP and MPVC Formulations for Solving Complex Nonlinear Decision-Making Problems in Aerospace Applications

Andrea Ghezzi, Armin Nurkanović, Avishai Weiss, Moritz Diehl, Stefano Di Cairano

TL;DR

This paper examines decision-making problems for nonlinear aerospace systems that involve conditional activations of constraints and objectives. It compares two reformulations—MINLP with discrete decisions and MPVC with continuous variables under vanishing constraints—applied to optimal trajectory planning, including powered descent guidance with divert-feasible regions. Through analysis and numerical case studies, it shows that MPVC can yield faster, accurate solutions for moderately nonconvex problems, but experiences convergence challenges as nonconvexity grows, whereas MINLP tends to be more robust and scalable for larger, more complex instances. The findings guide practitioners in selecting the formulation and solver strategy (e.g., S-B-MIQP for MINLP or homotopy for MPVC) based on problem size, structure, and available computational resources, with demonstrated implications for real-time aerospace decision-making.

Abstract

High-level decision-making for dynamical systems often involves performance and safety specifications that are activated or deactivated depending on conditions related to the system state and commands. Such decision-making problems can be naturally formulated as optimization problems where these conditional activations are regulated by discrete variables. However, solving these problems can be challenging numerically, even on powerful computing platforms, especially when the dynamics are nonlinear. In this work, we consider decision-making for nonlinear systems where certain constraints, as well as possible terms in the cost function, are activated or deactivated depending on the system state and commands. We show that these problems can be formulated either as mixed-integer nonlinear programs (MINLPs) or as mathematical programs with vanishing constraints (MPVCs), where the former formulation involves discrete decision variables, whereas the latter relies on continuous variables subject to structured nonconvex constraints. We discuss the different solution methods available for both formulations and demonstrate them on optimal trajectory planning problems in various aerospace applications. Finally, we compare the strengths and weaknesses of the MINLP and MPVC approaches through a focused case study on powered descent guidance with divert-feasible regions.

A Comparative Study of MINLP and MPVC Formulations for Solving Complex Nonlinear Decision-Making Problems in Aerospace Applications

TL;DR

This paper examines decision-making problems for nonlinear aerospace systems that involve conditional activations of constraints and objectives. It compares two reformulations—MINLP with discrete decisions and MPVC with continuous variables under vanishing constraints—applied to optimal trajectory planning, including powered descent guidance with divert-feasible regions. Through analysis and numerical case studies, it shows that MPVC can yield faster, accurate solutions for moderately nonconvex problems, but experiences convergence challenges as nonconvexity grows, whereas MINLP tends to be more robust and scalable for larger, more complex instances. The findings guide practitioners in selecting the formulation and solver strategy (e.g., S-B-MIQP for MINLP or homotopy for MPVC) based on problem size, structure, and available computational resources, with demonstrated implications for real-time aerospace decision-making.

Abstract

High-level decision-making for dynamical systems often involves performance and safety specifications that are activated or deactivated depending on conditions related to the system state and commands. Such decision-making problems can be naturally formulated as optimization problems where these conditional activations are regulated by discrete variables. However, solving these problems can be challenging numerically, even on powerful computing platforms, especially when the dynamics are nonlinear. In this work, we consider decision-making for nonlinear systems where certain constraints, as well as possible terms in the cost function, are activated or deactivated depending on the system state and commands. We show that these problems can be formulated either as mixed-integer nonlinear programs (MINLPs) or as mathematical programs with vanishing constraints (MPVCs), where the former formulation involves discrete decision variables, whereas the latter relies on continuous variables subject to structured nonconvex constraints. We discuss the different solution methods available for both formulations and demonstrate them on optimal trajectory planning problems in various aerospace applications. Finally, we compare the strengths and weaknesses of the MINLP and MPVC approaches through a focused case study on powered descent guidance with divert-feasible regions.

Paper Structure

This paper contains 26 sections, 4 theorems, 48 equations, 9 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Decision-making via optimal control
  3. Discrete-time optimal control problem
  4. Aerospace trajectory planning case studies
  5. Powered descent guidance with divert-feasible trajectories
  6. Coordination of unmanned ground and aerial vehicles
  7. Soft docking
  8. Abort-safe spacecraft rendezvous
  9. Obtaining computationally tractable logical expressions
  10. Comparison of the reformulations
  11. Explicit formulations for problems with Heaviside step-function and indicator constraints
  12. Methods for solving MINLP and MPVC
  13. Solving MINLPs
  14. Nonlinear branch-and-bound
  15. Sequential Benders-based MIQP
  16. ...and 11 more sections

Key Result

lemma 1

Let $(z^\star, \delta^\star)$ be a locally optimal solution of the relaxed MINLP op: generic minlp, i.e., with $\delta \in [0, 1]^{n_\delta}$, and let $\mathcal{I} \subseteq \mathbb{Z}_{[1, n_\delta]}$ be the set of indices such that $G_i(z^\star) \leq 0$ for all $i \in \mathcal{I}$, then $\delta_i^

Figures (9)

  • Figure 1: Left: in dark green the feasible set of constraint \ref{['cns: generic bigM constraint']} with $\delta_i \in \{0, 1\}$, and in light green its relaxation. i.e., $\delta_i \in [0, 1]$. Right: in light green the feasible set of constraint \ref{['cns: generic vanishing constraint']}.
  • Figure 2: Left: in light green the feasible set of constraint \ref{['eq: indicator constraint integer formulation']} (the indicator variable $\delta_i$ is projected out). Right: in light green the feasible set of constraint \ref{['eq: indicator constraint vanishing formulation']}. In both plots the dashed lines are part of the feasible set. The MPVC constraints \ref{['eq: indicator constraint vanishing formulation']} do not correctly represent the logical implication. By considering \ref{['eq: H(z) >= 0']}, we obtain two identical feasible sets and a correct representation of the logical implication. However, this introduces unnecessary restrictions for the MINLP formulation.
  • Figure 3: Feasible set of constraints \ref{['eq: eps indicator constraint integer formulation']}. The dashed lines are part of the feasible set, the red rectangle defined from zero to $\varepsilon$ is the portion of the feasible set removed.
  • Figure 4: Reformulations of the Heaviside step-function: in dashed orange the representation via the sigmoid function \ref{['eq: sigmoid fn as indicator function']}, in solid blue the representation via KKT conditions \ref{['eq: KKT representation of indicator function']}.
  • Figure 5: Locally optimal UGV position trajectories for MINLP (left) and MPVC (right) for \ref{['op: ugv OCP']}.
  • ...and 4 more figures

Theorems & Definitions (8)

  • lemma 1
  • proof
  • theorem 1
  • proof
  • lemma 2
  • lemma 3
  • remark 1
  • remark 2