Decomposition Algorithm Model for Singly Linearly-Constrained Problems Subject to Lower and Upper Bounds

• Published in: Journal of optimization theory and applications Vol. 141; no. 1; pp. 107 - 126
• Main Authors: Lin, C. J., Lucidi, S., Palagi, L., Risi, A., Sciandrone, M.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.04.2009; Springer; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Applied sciences; Calculus of Variations and Optimal Control; Optimization; Decomposition; Engineering; Exact sciences and technology; Mathematical programming; Mathematics; Mathematics and Statistics; Methods; Operational research and scientific management; Operational research. Management science; Operations Research/Decision Theory; Optimization; Studies; Support vector machines; Theory of Computation; Large scale optimization; Support vector machines; Decomposition methods; Freedom degree; Program optimization; Decomposition method; Modeling; Convex programming; Vector support machine; Numerical convergence; Large scale system; Lower bound; Objective analysis; Minimization; Quadratic programming; Global solution; Exact solution; Upper bound; Non differentiable programming; Equality constraint; Sparse set; Descent method; Problem solving; Large scale; Objective function
• DOI: 10.1007/s10957-008-9489-9
• ISSN: 0022-3239

Many real applications can be formulated as nonlinear minimization problems with a single linear equality constraint and box constraints. We are interested in solving problems where the number of variables is so huge that basic operations, such as the evaluation of the objective function or the updating of its gradient, are very time consuming. Thus, for the considered class of problems (including dense quadratic programs), traditional optimization methods cannot be applied directly. In this paper, we define a decomposition algorithm model which employs, at each iteration, a descent search direction selected among a suitable set of sparse feasible directions. The algorithm is characterized by an acceptance rule of the updated point which on the one hand permits to choose the variables to be modified with a certain degree of freedom and on the other hand does not require the exact solution of any subproblem. The global convergence of the algorithm model is proved by assuming that the objective function is continuously differentiable and that the points of the level set have at least one component strictly between the lower and upper bounds. Numerical results on large-scale quadratic problems arising in the training of support vector machines show the effectiveness of an implemented decomposition scheme derived from the general algorithm model.