A new polynomial-time algorithm for linear programming
Combinatorica
A "build-down" scheme for linear programming
Mathematical Programming: Series A and B
An OL(n3) potential reduction algorithm for linear programming
Mathematical Programming: Series A and B
An active-set strategy in an interior point method for linear programming
Mathematical Programming: Series A and B
Machine Learning
Primal-dual interior-point methods
Primal-dual interior-point methods
Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
An Analytic Center Based Column Generation Algorithm for Convex Quadratic Feasibility Problems
SIAM Journal on Optimization
A Tutorial on Support Vector Machines for Pattern Recognition
Data Mining and Knowledge Discovery
Interior-Point Methods for Massive Support Vector Machines
SIAM Journal on Optimization
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Constraint Reduction for Linear Programs with Many Inequality Constraints
SIAM Journal on Optimization
Newton-KKT interior-point methods for indefinite quadratic programming
Computational Optimization and Applications
Adaptive constraint reduction for convex quadratic programming and training support vector machines
Adaptive constraint reduction for convex quadratic programming and training support vector machines
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We propose an adaptive, constraint-reduced, primal-dual interior-point algorithm for convex quadratic programming with many more inequality constraints than variables. We reduce the computational effort by assembling, instead of the exact normal-equation matrix, an approximate matrix from a well chosen index set which includes indices of constraints that seem to be most critical. Starting with a large portion of the constraints, our proposed scheme excludes more unnecessary constraints at later iterations. We provide proofs for the global convergence and the quadratic local convergence rate of an affine-scaling variant. Numerical experiments on random problems, on a data-fitting problem, and on a problem in array pattern synthesis show the effectiveness of the constraint reduction in decreasing the time per iteration without significantly affecting the number of iterations. We note that a similar constraint-reduction approach can be applied to algorithms of Mehrotra's predictor-corrector type, although no convergence theory is supplied.