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<p>In this article, a novel Three-phase method is proposed to systematically and effectively compute multiple local optimal solutions (LOSs) for nonlinear constrained optimization problems. It locates the feasible connected component (FCC) by reformulating the problem as the identification of the stable equilibrium manifold (SEM) of a Quotient Gradient System (QGS) and transforms the search for a local optimal solution into the identification of stable equilibrium points (SEP) of a Projected Gradient System (PGS). In phase I, a FCC is identified by following the QGS; phase II enables transition from the current FCC to others; in phase III, the PGS is applied to find the LOSs simultaneously on the FCCs. This method offers systematic and simultaneous exploration of multiple solutions to obtain the high-quality solution. A theoretical foundation for each phase is developed, and these theoretical results are quite general on their own. The proposed method is numerically evaluated to compute multiple LOSs, for instance, a step-by-step demonstration using a 2-D example. In addition, compared with the interior point method, the heuristic method, and other multiple solution techniques, the proposed method can efficiently find multiple or even global optimal solutions in various real-world optimal power system and robotic manipulator problems.</p>

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A three-phase method for systematically and simultaneously computing multiple optimal solutions

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • Cuiqing Zhang,
  • Yusi Zhang

摘要

In this article, a novel Three-phase method is proposed to systematically and effectively compute multiple local optimal solutions (LOSs) for nonlinear constrained optimization problems. It locates the feasible connected component (FCC) by reformulating the problem as the identification of the stable equilibrium manifold (SEM) of a Quotient Gradient System (QGS) and transforms the search for a local optimal solution into the identification of stable equilibrium points (SEP) of a Projected Gradient System (PGS). In phase I, a FCC is identified by following the QGS; phase II enables transition from the current FCC to others; in phase III, the PGS is applied to find the LOSs simultaneously on the FCCs. This method offers systematic and simultaneous exploration of multiple solutions to obtain the high-quality solution. A theoretical foundation for each phase is developed, and these theoretical results are quite general on their own. The proposed method is numerically evaluated to compute multiple LOSs, for instance, a step-by-step demonstration using a 2-D example. In addition, compared with the interior point method, the heuristic method, and other multiple solution techniques, the proposed method can efficiently find multiple or even global optimal solutions in various real-world optimal power system and robotic manipulator problems.