scholarly journals Nonsingularity Conditions for FB System of Reformulating Nonlinear Second-Order Cone Programming

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Shaohua Pan ◽  
Shujun Bi ◽  
Jein-Shan Chen
Keyword(s):  
Constraint Qualification ◽  
Optimal Solution ◽  
Second Order ◽  
Strong Regularity ◽  
Cone Programming ◽  
Second Order Cone Programming ◽  
Second Order Cone ◽  
Constraint Nondegeneracy ◽  
Robinson’S Constraint Qualification ◽  
Locally Optimal Solution

This paper is a counterpart of Bi et al., 2011. For a locally optimal solution to the nonlinear second-order cone programming (SOCP), specifically, under Robinson’s constraint qualification, we establish the equivalence among the following three conditions: the nonsingularity of Clarke’s Jacobian of Fischer-Burmeister (FB) nonsmooth system for the Karush-Kuhn-Tucker conditions, the strong second-order sufficient condition and constraint nondegeneracy, and the strong regularity of the Karush-Kuhn-Tucker point.

scholarly journals Quantum algorithms for Second-Order Cone Programming and Support Vector Machines

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 427
Author(s):  
Iordanis Kerenidis ◽  
Anupam Prakash ◽  
Dániel Szilágyi
Keyword(s):  
Quantum Algorithms ◽  
Matrix Multiplication ◽  
Optimal Solution ◽  
Quantum Algorithm ◽  
Second Order ◽  
Support Vector ◽  
Scaling Exponent ◽  
Cone Programming ◽  
Second Order Cone Programming ◽  
Second Order Cone

We present a quantum interior-point method (IPM) for second-order cone programming (SOCP) that runs in time O~(nrζκδ2log⁡(1/ϵ)) where r is the rank and n the dimension of the SOCP, δ bounds the distance of intermediate solutions from the cone boundary, ζ is a parameter upper bounded by n, and κ is an upper bound on the condition number of matrices arising in the classical IPM for SOCP. The algorithm takes as its input a suitable quantum description of an arbitrary SOCP and outputs a classical description of a δ-approximate ϵ-optimal solution of the given problem.Furthermore, we perform numerical simulations to determine the values of the aforementioned parameters when solving the SOCP up to a fixed precision ϵ. We present experimental evidence that in this case our quantum algorithm exhibits a polynomial speedup over the best classical algorithms for solving general SOCPs that run in time O(nω+0.5) (here, ω is the matrix multiplication exponent, with a value of roughly 2.37 in theory, and up to 3 in practice). For the case of random SVM (support vector machine) instances of size O(n), the quantum algorithm scales as O(nk), where the exponent k is estimated to be 2.59 using a least-squares power law. On the same family random instances, the estimated scaling exponent for an external SOCP solver is 3.31 while that for a state-of-the-art SVM solver is 3.11.

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