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<p>This study focuses on a class of generalized linear multiplicative programming problems arising in complex fields such as venture capital analysis and system stability assessment. We propose a global optimization algorithm that integrates a nonlinear convex relaxation (CR), a linear approximation method, and a simplicial branching strategy. First, an equivalent problem is constructed to project the original problem onto a lower-dimensional space. By utilizing the convex envelope properties of the concave component of the objective function over a simplex domain, a new CR technique is developed to construct a convex underestimator of the nonconvex objective. Subsequently, a linear approximation scheme is designed to locate the KKT points of the equivalent problem. By embedding the proposed CR method and simplicial branching rule within a branch-and-bound framework, an innovative algorithm with guaranteed global convergence is developed. Theoretical analysis shows that for any given tolerance <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the algorithm converges to a global <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-optimal solution of the equivalent problem, and an estimate of its computational complexity is provided. Numerical results validate the effectiveness and feasibility of the proposed algorithm.</p>

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Global Optimization for Generalized Linear Multiplicative Programs Through Simplicial Branch and Bound with Convex Relaxation

  • 期刊论文

  • 发表日期：

    2026年1月25日

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  • Bo Zhang,
  • Kai Cao,
  • Yue-Lin Gao

摘要

This study focuses on a class of generalized linear multiplicative programming problems arising in complex fields such as venture capital analysis and system stability assessment. We propose a global optimization algorithm that integrates a nonlinear convex relaxation (CR), a linear approximation method, and a simplicial branching strategy. First, an equivalent problem is constructed to project the original problem onto a lower-dimensional space. By utilizing the convex envelope properties of the concave component of the objective function over a simplex domain, a new CR technique is developed to construct a convex underestimator of the nonconvex objective. Subsequently, a linear approximation scheme is designed to locate the KKT points of the equivalent problem. By embedding the proposed CR method and simplicial branching rule within a branch-and-bound framework, an innovative algorithm with guaranteed global convergence is developed. Theoretical analysis shows that for any given tolerance \(\varepsilon >0\) ε > 0 , the algorithm converges to a global \(\varepsilon \) ε -optimal solution of the equivalent problem, and an estimate of its computational complexity is provided. Numerical results validate the effectiveness and feasibility of the proposed algorithm.