A cutting-plane method for Mixed-Logical Semidefinite Programs with an application to multi-vehicle robust path planning

Abstract

The usual approach to dealing with Mixed Logical Semidefinite Programs (MLSDPs) is through the "Big-M" or the convex hull reformulation. The Big-M approach is appealing for its ease of modeling, but it leads to weak convex relaxations when used in a Branch & Bound framework. The convex hull reformulation, on the other hand, introduces a significant number of auxiliary variables and constraints and is only applicable if the feasible region consists of several disjunctive bounded polyhedra. This paper aims to circumvent these shortcomings by leveraging on Combinatorial Benders Cuts due to Codato & Fischetti and by constructing linear cuts based on a Farkas Lemma for Semidefinite Programming (SDP) within a Cutting-Plane framework. We employ the resulting Cutting-Plane algorithm in a Robust Model Predictive Control (RMPC) test application for multi-vehicle robust path planning with obstacle and inter-vehicle collision avoidance, taking into consideration exogenous (eg external wind gusts) and endogenous (eg internal noise in the system gain) uncertainty. We formulate this problem as an MLSDP model using minimax approaches by Löfberg and by El Ghaoui et al. and Big-M formulations due to Richards & How.

Algorithms for joint sensor and control nodes selection in dynamic networks

“…The proof of Theorem 1 is given in Appendix C. This theorem allows the SSASP to be solved as a MI-SDP, which can be handled using a variety of optimization methods such as: branch-and-bound algorithms [10,11], outer approximations [20], or cutting-plane methods [15]. The next section presents a departure from MI-SDP to an algorithm that returns optimal solutions to SSASP, without requiring L 1,2,3 .…”

“…Instead, we can learn the feasibility of specific SA selection using these algebraic conditions. Finally, and instead of solving the MI-SDP form of SSASP using the classical BnB algorithm, we plan to test the performance of the outer approximations [20] and the cutting-plane methods [15]-as these approaches have shown significant savings for the computational time especially when compared with classical BnB methods.…”

Combining qualitative information and semi‐quantitative data for guaranteed invalidation of biochemical network models

“…For a more detailed discussion of the involved relaxation and reformulation steps, we refer to [34,36]. Note that the MISDP can also be formulated as an optimization problem, for example, [30,32,37], which allows the use of more elaborate algorithms to derive an outer-approximation of P .…”

“…Although, some recent reports suggest that there will be methods for deriving valid cuts for MISDP (e.g., [37]), there are yet no efficient solvers available. We relax therefore MISDP further into a MILP .…”

Time-Varying Sensor and Actuator Selection for Uncertain Cyber-Physical Systems

“…More elaborate quantitative notions based on Gramians [6]- [14] and classical optimal and robust control Ahmad F. Taha and estimation problems [15]- [22] for linear systems have also been studied. For selecting SaAs based on these metrics, several optimization methods are proposed in this literature, including combinatorial greedy algorithms [8], [9], [19], [21], [23], convex relaxation heuristics using sparsity-inducing ℓ 1 penalty functions [15]- [18] and reformulations to mixedinteger semidefinite programming via the big-M method or McCormick's relaxation [13], [22], [24]. As a departure from control-theoretic frameworks, the authors in [25] explore an optimization-based method for reconstructing the initial states of nonlinear dynamic systems given (a) arbitrary nonlinear model, while (b) optimally selecting a fixed number of sensors.…”