{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T13:31:20Z","timestamp":1776691880137,"version":"3.51.2"},"reference-count":100,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T00:00:00Z","timestamp":1766361600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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\u2014a 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] \u00d7 [0, 1] \u2282 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\u2013Kuhn\u2013Tucker 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\u2014avoiding zig-zag trajectories\u2014and reach (0, 1) or (1, 0), where both feasibility and optimality are satisfied at once.<\/jats:p>","DOI":"10.3390\/a19010011","type":"journal-article","created":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T16:19:26Z","timestamp":1766420366000},"page":"11","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Avoiding the Maratos Effect in Non-Convex Optimization Through Piecewise Convexity: A Case Study in Optimal PMU Placement Problem"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0666-9544","authenticated-orcid":false,"given":"Nikolaos P.","family":"Theodorakatos","sequence":"first","affiliation":[{"name":"School of Electrical & Computer Engineering, National Technical University of Athens, 157 80 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4128-4428","authenticated-orcid":false,"given":"Rohit","family":"Babu","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Madan Mohan Malaviya University of Technology, Gorakhpur 273010, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7281-5458","authenticated-orcid":false,"given":"Miltiadis D.","family":"Lytras","sequence":"additional","affiliation":[{"name":"Computer Science Department, College of Engineering, Effat University, Jeddah 21478, Saudi Arabia"},{"name":"Department of Management, School of Business and Economics, The American College of Greece, 153 42 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Luenberger, D.G., and Yonge, Y. 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