几何规划与符号规划的MM算法
MM Algorithms for Geometric and Signomial Programming
AI总结：
本文基于MM算法框架，利用几何-算术均值不等式和支持超平面不等式，将无约束符号规划转化为一系列一维最小化问题，并扩展至约束情形，特别适用于二次规划。
摘要：
本文推导了符号规划（几何规划的推广）的新算法。这些算法基于称为MM算法的通用优化原理。在此设置中，可以应用几何-算术均值不等式和支持超平面不等式来创建参数分离的替代函数。因此，无约束符号规划简化为一系列一维最小化问题。简单例子表明，推导出的MM算法可以收敛到边界点或连续最小点集合中的一点。对于几何规划，证明了最小点唯一或出现在参数空间内部的条件。收敛到内点以线性速率发生。最后，MM框架容易处理符号类型的等式和不等式约束。对于最重要的特例——约束二次规划，MM算法涉及非常简单的更新。
英文摘要：
This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming reduces to a sequence of one-dimensional minimization problems. Simple examples demonstrate that the MM algorithm derived can converge to a boundary point or to one point of a continuum of minimum points. Conditions under which the minimum point is unique or occurs in the interior of parameter space are proved for geometric programming. Convergence to an interior point occurs at a linear rate. Finally, the MM framework easily accommodates equality and inequality constraints of signomial type. For the most important special case, constrained quadratic programming, the MM algorithm involves very simple updates.
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