Disjunctive Programming and Piecewise Convexity: An Algorithmic Trajectory Analysis Toward Stationary Points to Avoid the Maratos Effect in Numerical Optimization

Abstract

In this paper, we present an algorithmic modeling approach based on Disjunctive Programming using Boolean logic “OR” to solve the Optimal Phasor Measurement Unit Placement (OPP). We propose a framework for modeling the optimal PMU placement subject to disjunctive constraints. A convex objective function is minimized subject to a bilinear equality constraint with a piecewise linear structure. The polynomial constraint constitutes a union of linear segments conceptually analyzed in the two-dimensional continuous space, separating the infeasible from the feasible region. This work investigates the trajectory of iterates from infeasible initial points to stationary solutions and analyzes the convergence behavior using Interior-Point Method (IPM) and Sequential Quadratic Programming (SQP). Our algorithmic model addresses the progress of infeasible and feasible iterates, step computation using line-search and trust-region mechanisms. Combined with second-order correction (SOC) and filter methods, these mechanisms enable the algorithm to maintain a unit primal step, even when starting from an infeasible initial point. Network observability constraints are transformed from a Conjunctive Normal Form (CNF) into a Disjunctive Normal Form (DNF) via Balas’s theory. This geometry transformation reformulates the feasible set into a union of convex affine pieces, effectively eliminating constraint curvature issues. This affine reformulation ensures that gradient-based algorithms maintain a smooth optimization trajectory along a convex local manifold. This trajectory enables the algorithm to preserve the full Newton step, maintaining a superlinear convergence rate. Its underlying piecewise linear convexity inherently enables the gradient-based algorithm to avoid the Maratos effect. Numerical results on IEEE power systems validate the optimization problem. Our framework uses the IEEE-14 bus system to address the high-degree non-convexities. Monte Carlo simulations further enhance the argument that piecewise convexity enables IPM and SQP to converge to binary local minima. Depending on multiple-run initialization, these methods reach the same objective function value. These local minima can be characterized as non-strict optimum points that are structurally symmetric but exhibit unequal basins of attraction. Hence, this geometry-driven formulation enables gradient-based methods to reliably identify binary-valued optimal solutions for the OPP.