Approximate Duality

• Published in: Journal of optimization theory and applications Vol. 135; no. 3; pp. 429 - 443
• Main Authors: SCOVEL, C, HUSH, D, STEINWART, I
• Format: Journal Article
• Language: English
• Published: New York, NY Springer 01.12.2007; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Applied sciences; Approximation; Convex analysis; Exact sciences and technology; Lagrange multiplier; Learning; Learning theory; Mathematical analysis; Mathematical programming; Operational research and scientific management; Operational research. Management science; Optimization; Quadratic programming; Saddle points; Studies; Support vector machines; Theory; Vectors (mathematics); Saddle point method; Lagrangian duality; Lagrangian; Subgradient optimization; Saddle points; Lagrangian theory; Approximations; Duality; Quadratic programming; Convex programming; Saddle point; Kuhn-Tucker conditions, Support vector machines; Lagrange multiplier; Vector support machine; Kuhn Tucker condition; Kuhn Tucker method
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-007-9281-2

We extend the Lagrangian duality theory for convex optimization problems to incorporate approximate solutions. In particular, we generalize well-known relationships between minimizers of a convex optimization problem, maximizers of its Lagrangian dual, saddle points of the Lagrangian, Kuhn-Tucker vectors, and Kuhn-Tucker conditions to incorporate approximate versions. As an application, we show how the theory can be used for convex quadratic programming and then apply the results to support vector machines from learning theory.