Sensitivity-Based Distributed Programming for Non-Convex Optimization

TL;DR

This work introduces Sensitivity-Based Distributed Programming (SBDP) for large-scale, non-convex NLPs by integrating first-order neighbor sensitivities into local subproblems and solving them in parallel under primal decomposition. It develops two variants—general SBDP and a neighbor-affine special case—that reduce communication and computation, and it establishes local convergence guarantees via a Fiacco-style sensitivity framework, with the Jacobian J(p) = -M(p)^{-1} N(p) governing contraction. The method is extended to distributed optimal control, including a DMPC formulation, and convergence can be enforced by choosing an appropriate prediction horizon T (with T_max bounding convergence). Numerical experiments on non-convex NLPs, coupled inverted pendulums, and a smart-grid DMPC scenario validate the theory, show linear convergence in general and potential quadratic convergence in special cases, and demonstrate scalability with network size and neighbor structure.

Abstract

This paper presents a novel sensitivity-based distributed programming (SBDP) approach for non-convex, large-scale nonlinear programs (NLP). The algorithm relies on first-order sensitivities to cooperatively solve the central NLP in a distributed manner with only neighbor-to-neighbor communication and parallelizable local computations. The scheme is based on primal decomposition and offers minimal algorithmic complexity. We derive sufficient local convergence conditions for non-convex problems. Furthermore, we consider the SBDP method in a distributed optimal control context and derive favorable convergence properties in this setting. We illustrate these theoretical findings and the performance of the proposed algorithm with simulations of various distributed optimization and control problems.

Figures (5)

• Figure 1: Contour lines of NLP \ref{['eq:example_NLP']} and iteration profiles of Algorithm \ref{['alg:SENSI_with_neighboraffinity']}. The initializations of the blue iterates lie within $\mathcal{B}_r(\boldsymbol{x}^{*})$, while those in red do not. However, the SBDP method still converges for the initializations outside $\mathcal{B}_r(\boldsymbol{x}^{*})$ as the practical convergence radius is typically much larger.
• Figure 2: Error progression for different initial guesses (left) and observed convergence rate $C^q$ (right) of Algorithm \ref{['alg:SENSI_with_neighboraffinity']} for NLP \ref{['eq:example_NLP']}. The dashed lines show the norm of the Jacobian, $\|J(\boldsymbol{x}^{*|n})\|$, at the respective local minima $n$.
• Figure 3: Trajectories of agents $1$, $5$ and $10$ for the side-stepping of multiple coupled inverted pendulums for $m_c = 1\, kg$, $m_l=0.25\, kg$$l=0.5\, m$ and $c=0.25\, N \per m$. The dashed lines indicate the respective constraints.
• Figure 4: Error progression of Algorithm \ref{['alg:SENSI_with_neighboraffinity']} for NLP \ref{['eq:discrete_time_OCP']} for a varying spring stiffness $c>0$.
• Figure 5: Behavior of the norm of the Jacobian $\| \boldsymbol{J}(\boldsymbol{p}^{*})\|$ w.r.t. to the prediction horizon length $T$ in dependence of different parameter values. In each figure, the value of only one parameter is varied while the other parameters are kept constant at their default value at $I=0.2$, $A=0.1$, $\boldsymbol{Q}=\mathop{\mathrm{\mathrm{diag}}}\limits(0,1)$ and $\mathcal{G}_1$. The intersection of the respective curves with the dashed lines, i.e., where $\| \boldsymbol{J}(\boldsymbol{p}^{*})\| = 1$, indicate the maximum allowable prediction horizon length $T_{\mathrm{max}}$.

Theorems & Definitions (17)

• Remark 1
• Example 1
• Lemma 1
• Lemma 2
• Lemma 3
• Remark 2
• Example 1: cont.
• Theorem 1
• Remark 3
• Corollary 1