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Quadratic convergence of Newton's method to the optimal solution of second-order conic optimization

Ali Mohammad-Nezhad(alm413***at***lehigh.edu)
Tamas Terlaky(terlaky***at***lehigh.edu)

Abstract: Under strict complementarity and primal and dual nondegeneracy conditions we establish the quadratic convergence of Newton's method to the unique strictly complementary optimal solution of second-order conic optimization, when the initial point is sufficiently close to the optimal set. When strict complementarity fails but the primal and dual nondegeneracy conditions hold, we show that if the optimal partition of the problem is known, then the application of Newton's method to the optimality conditions of a reduced nonlinear optimization problem results in quadratic convergence to the unique maximally complementary optimal solution. For a special case of the optimal partition, we present a rounding procedure which gives an exact strictly complementary optimal solution in strongly polynomial time.

Keywords: Second-order conic optimization, optimal partition, maximally complementary optimal solution, nondegeneracy conditions, second-order sufficient condition

Category 1: Linear, Cone and Semidefinite Programming (Second-Order Cone Programming )

Citation: Report 17T-014, Industrial and Systems Engineering, Lehigh University, Oct. 2017

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Entry Submitted: 10/26/2017
Entry Accepted: 10/26/2017
Entry Last Modified: 10/26/2017

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