Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs
Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs
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DOI：
10.1023/b:jota.0000043992.38554.c8
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发表时间：
2004-10
影响因子：
1.9
通讯作者：
V. Jeyakumar;G. M. Lee;Nguyen Nang Dinh
中科院分区：
文献类型：
--
作者：
V. Jeyakumar;G. M. Lee;Nguyen Nang Dinh
给出了锥约束凸优化问题最优解集的各种表征。结果以次梯度和拉格朗日乘数表示。我们首先确定凸规划的拉格朗日函数在最优解集上是常数。然后使用该基本属性导出解集的各种简单的基于拉格朗日乘子的表征。对于具有不等式约束的有限维凸程序，其特征表明，最优解处具有正拉格朗日乘数的主动约束在程序的所有最优解处均保持有效。结果应用于导出半定程序和分数程序的解集的相应拉格朗日乘子特征。给出了具体例子来说明结果的性质。
Various characterizations of optimal solution sets of cone-constrained convex optimization problems are given. The results are expressed in terms of subgradients and Lagrange multipliers. We establish first that the Lagrangian function of a convex program is constant on the optimal solution set. This elementary property is then used to derive various simple Lagrange multiplier-based characterizations of the solution set. For a finite-dimensional convex program with inequality constraints, the characterizations illustrate that the active constraints with positive Lagrange multipliers at an optimal solution remain active at all optimal solutions of the program. The results are applied to derive corresponding Lagrange multiplier characterizations of the solution sets of semidefinite programs and fractional programs. Specific examples are given to illustrate the nature of the results.