Nonlinear Lagrangian Theory for Nonconvex Optimization

C. J. Goh and X. Q. Yang

Journal of Optimization Theory and Applications, 2001, vol. 109, issue 1, No 6, 99-121

Abstract: Abstract The Lagrangian function in the conventional theory for solving constrained optimization problems is a linear combination of the cost and constraint functions. Typically, the optimality conditions based on linear Lagrangian theory are either necessary or sufficient, but not both unless the underlying cost and constraint functions are also convex. We propose a somewhat different approach for solving a nonconvex inequality constrained optimization problem based on a nonlinear Lagrangian function. This leads to optimality conditions which are both sufficient and necessary, without any convexity assumption. Subsequently, under appropriate assumptions, the optimality conditions derived from the new nonlinear Lagrangian approach are used to obtain an equivalent root-finding problem. By appropriately defining a dual optimization problem and an alternative dual problem, we show that zero duality gap will hold always regardless of convexity, contrary to the case of linear Lagrangian duality.

Keywords: inequality constraints; nonlinear Lagrangian; nonconvex optimization; sufficient and necessary conditions; zero duality gap (search for similar items in EconPapers)
Date: 2001
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Citations: View citations in EconPapers (7)

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DOI: 10.1023/A:1017513905271

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