Avoiding the Maratos Effect in Non-Convex Optimization Through Piecewise Convexity: A Case Study in Optimal PMU Placement Problem

In constrained nonlinear optimization, we aim to achieve two goals: one is to minimize the objective function, and the other is to satisfy the constraints. A common way to balance these competing targets is to use penalty functions. Suppose that an algorithm generates a descent direction and produces a step that decreases the objective function value but increases the constraint violation—a phenomenon known as the Maratos effect. This leads to the rejection of the full step by the non-smooth penalty function; therefore, superlinear convergence is not preserved. This work leverages a piecewise convexity model to solve the optimal PMU placement. A quadratic objective function is minimized subject to a non-convex equality constraint within box constraints [0, 1] × [0, 1] ⊂ R2. The initial non-convex region is reconsidered as a union of piecewise line segments. This decomposition enables algorithms to converge to a local optimum while preserving superlinear convergence near the solution. An analytical solution is presented using the Karush–Kuhn–Tucker conditions. First-and-second-order optimality conditions are applied to find the local minimum. We show how the Maratos effect is avoided by adopting the piecewise convexity without needing a non-smooth penalty function, second-order corrections or employing the watchdog methods. Simulations demonstrate that the algorithms partially search the space along the line segments—avoiding zig-zag trajectories—and reach (0, 1) or (1, 0), where both feasibility and optimality are satisfied at once.