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June  2020, 10(2): 165-175. doi: 10.3934/naco.2019046

## Quadratic optimization with two ball constraints

 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

* Corresponding author: Maziar Salahi

Received  December 2018 Revised  July 2019 Published  September 2019

Fund Project: The authors are grateful to the reviewers for their comments and suggestions and University of Guilan for partially supporting this research

In this paper, the minimization of a general quadratic function subject to two ball constraints, called two ball trust-region subproblem (TBTRS), is studied. It is shown that the global optimal solution can be found by solving two extended trust-region subproblems. Strong duality conditions for two special cases are discussed. Finally, a comparison of results of the new algorithm with the other two recently proposed algorithms and CVX software are presented for several classes of randomly generated test problems.

Citation: Saeid Ansary Karbasy, Maziar Salahi. Quadratic optimization with two ball constraints. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 165-175. doi: 10.3934/naco.2019046
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##### References:
$+$: global solution of $Pc_1$ and $Pc_2$, $\Box$: LNGMs of $Pc_1$ and $Pc_2$
Notations in the tables
 Notation Description n Dimension of problem Den Density of A CPU Run time $F_{TB}$ Objective value of the new algorithm $F_{AD}$ Objective value of ADMM algorithm [2] $F_{SD}$ Objective value of SDP relaxation $F_{SA}$ Objective value of Sakaue et. al's algorithm [25] $x_{TB}^*$ Optimal solution of the new algorithm $x_{SA}^*$ Optimal solution of Sakaue et. al's algorithm [25]
 Notation Description n Dimension of problem Den Density of A CPU Run time $F_{TB}$ Objective value of the new algorithm $F_{AD}$ Objective value of ADMM algorithm [2] $F_{SD}$ Objective value of SDP relaxation $F_{SA}$ Objective value of Sakaue et. al's algorithm [25] $x_{TB}^*$ Optimal solution of the new algorithm $x_{SA}^*$ Optimal solution of Sakaue et. al's algorithm [25]
Comparison of the new algorithm with ADMM algorithm [2] (first class)
 n Den CPU(TBTRS) CPU(ADMM) $F_{TB}-F_{AD}$ 50 1 0.23 0.36 -9.31e-11 100 1 0.28 0.47 -1.39 e-10 200 1 0.55 0.79 2.33e-11 300 1 0.80 1.75 1.86 e-10 400 1 1.58 2.63 4.66e-10 500 1 2.72 3.99 8.73e-10 700 0.1 1.49 5.174 4.54e-10 1000 0.1 3.27 20.64 3.49e-10 2000 0.01 2.48 7.64 2.79e-10 5000 0.001 2.74 6.59 -1.16e-10 10000 0.0001 4.62 7.29 -1.46e-11
 n Den CPU(TBTRS) CPU(ADMM) $F_{TB}-F_{AD}$ 50 1 0.23 0.36 -9.31e-11 100 1 0.28 0.47 -1.39 e-10 200 1 0.55 0.79 2.33e-11 300 1 0.80 1.75 1.86 e-10 400 1 1.58 2.63 4.66e-10 500 1 2.72 3.99 8.73e-10 700 0.1 1.49 5.174 4.54e-10 1000 0.1 3.27 20.64 3.49e-10 2000 0.01 2.48 7.64 2.79e-10 5000 0.001 2.74 6.59 -1.16e-10 10000 0.0001 4.62 7.29 -1.46e-11
Comparison of the new algorithm with Sakaue et. al's algorithm [25] (first class)
 n CPU(TBTRS) CPU(Sakaue et. al's algorithm [25]) $F_{TB}-F_{SA}$ $||x^*_{TB}-x^*_{SA}||$ 5 0.10 0.07 -2.27e-14 3.24e-14 10 0.12 0.46 -1.71e-11 1.45e-13 15 0.23 5.94 -3.87e-12 1.78e-14 20 0.18 33.12 -1.82e-09 8.06e-08 25 0.23 130.96 -2.96e-11 9.50e-14 30 0.22 428.21 -6.58e-09 1.26e-10
 n CPU(TBTRS) CPU(Sakaue et. al's algorithm [25]) $F_{TB}-F_{SA}$ $||x^*_{TB}-x^*_{SA}||$ 5 0.10 0.07 -2.27e-14 3.24e-14 10 0.12 0.46 -1.71e-11 1.45e-13 15 0.23 5.94 -3.87e-12 1.78e-14 20 0.18 33.12 -1.82e-09 8.06e-08 25 0.23 130.96 -2.96e-11 9.50e-14 30 0.22 428.21 -6.58e-09 1.26e-10
Comparison of the new algorithm with ADMM algorithm [2] (second class)
 n Den CPU(TBTRS) CPU(ADMM) $F_{TB}-F_{AD}$ 50 1 0.091 2.46 -3.69e-08 100 1 0.12 3.0022 -1.63e-07 200 1 0.15 4.70 -3.81e-07 300 1 0.24 6.71 -4.61e-07 500 1 0.68 13.5 -6.15e-07 700 1 1.17 21.43 -2.05e-06 1000 1 2.82 47.35 -1.69e-06 2000 0.1 2.39 39.16 -9.22e-07 5000 0.01 2.96 55.92 -1.61e-06 10000 0.001 3.14 132.33 -4.85e-06
 n Den CPU(TBTRS) CPU(ADMM) $F_{TB}-F_{AD}$ 50 1 0.091 2.46 -3.69e-08 100 1 0.12 3.0022 -1.63e-07 200 1 0.15 4.70 -3.81e-07 300 1 0.24 6.71 -4.61e-07 500 1 0.68 13.5 -6.15e-07 700 1 1.17 21.43 -2.05e-06 1000 1 2.82 47.35 -1.69e-06 2000 0.1 2.39 39.16 -9.22e-07 5000 0.01 2.96 55.92 -1.61e-06 10000 0.001 3.14 132.33 -4.85e-06
Comparison of the new algorithm with Sakaue et. al's algorithm [25] (second class)
 n CPU(TBTRS) CPU(Sakaue et. al's algorithm [25]) $F_{TB}-F_{SA}$ $||x^*_{TB}-x^*_{SA}||$ 5 0.06 0.051 1.42e-11 1.22e-06 10 0.06 0.42 2.58e-10 3.69e-06 15 0.12 6.18 -1.72e-10 2.89e-06 20 0.08 35.03 2.84e-09 1.07e-05 25 0.13 139.93 2.96e-09 1.02e-05 30 0.14 449.08 4.73e-08 3.45e-05
 n CPU(TBTRS) CPU(Sakaue et. al's algorithm [25]) $F_{TB}-F_{SA}$ $||x^*_{TB}-x^*_{SA}||$ 5 0.06 0.051 1.42e-11 1.22e-06 10 0.06 0.42 2.58e-10 3.69e-06 15 0.12 6.18 -1.72e-10 2.89e-06 20 0.08 35.03 2.84e-09 1.07e-05 25 0.13 139.93 2.96e-09 1.02e-05 30 0.14 449.08 4.73e-08 3.45e-05
Comparison of the new algorithm with SDP relaxation (third class)
 n Den CPU(TBTRS) CPU(ADMM) CPU(CVX) $F_{TB}-F_{AD}$ $F_{TB}-F_{SD}$ 50 1 0.09 2.04 0.97 2.30e-09 -1.58e-05 100 1 0.18 2.01 2.93 1.75e-08 -7.33e-05 200 1 0.16 4.05 5.41 3.82e-07 -9.45e-04 300 1 0.24 6.16 14.39 5.71e-07 -2.23e-4 400 1 0.41 10.78 38.06 1.35e-06 -3.78e-4 500 1 0.67 11.87 63.26 4.67e-07 -4.67e-4 700 0.1 0.32 8.39 132.96 1.43e-07 -2.75e-4 800 0.1 0.39 9.02 206.75 2.42e-07 -2.904e-4 900 0.1 0.44 9.67 268.47 8.91e-07 -4.88e-4 1000 0.1 0.52 11.21 427.06 3.29e-07 -5.71e-4 2000 0.1 2.69 39.09 $-$ 5.28e-05 $-$ 3000 0.1 5.27 82.52 $-$ 3.85e-05 $-$ 5000 0.01 2.99 57.99 $-$ 6.16e-05 $-$ 10000 0.001 2.34 121.13 $-$ 7.12e-05 $-$
 n Den CPU(TBTRS) CPU(ADMM) CPU(CVX) $F_{TB}-F_{AD}$ $F_{TB}-F_{SD}$ 50 1 0.09 2.04 0.97 2.30e-09 -1.58e-05 100 1 0.18 2.01 2.93 1.75e-08 -7.33e-05 200 1 0.16 4.05 5.41 3.82e-07 -9.45e-04 300 1 0.24 6.16 14.39 5.71e-07 -2.23e-4 400 1 0.41 10.78 38.06 1.35e-06 -3.78e-4 500 1 0.67 11.87 63.26 4.67e-07 -4.67e-4 700 0.1 0.32 8.39 132.96 1.43e-07 -2.75e-4 800 0.1 0.39 9.02 206.75 2.42e-07 -2.904e-4 900 0.1 0.44 9.67 268.47 8.91e-07 -4.88e-4 1000 0.1 0.52 11.21 427.06 3.29e-07 -5.71e-4 2000 0.1 2.69 39.09 $-$ 5.28e-05 $-$ 3000 0.1 5.27 82.52 $-$ 3.85e-05 $-$ 5000 0.01 2.99 57.99 $-$ 6.16e-05 $-$ 10000 0.001 2.34 121.13 $-$ 7.12e-05 $-$
Comparison of the new algorithm with Sakaue et. al's algorithm [25] (third class)
 n CPU(TBTRS) CPU(Sakaue et. al's algorithm [25]) $F_{AD}-F_{SA}$ $||x^*_{AD}-x^*_{SA}||$ 5 0.08 0.059 1.32e-13 9.59e-08 10 0.06 0.36 1.86e-09 1.08e-05 15 0.06 5.69 5.12e-09 1.66e-05 20 0.07 32.62 1.25e-09 6.09e-06 25 0.14 130.46 7.24e-10 3.75e-06 30 0.14 419.58 8.38e-09 1.72e-05
 n CPU(TBTRS) CPU(Sakaue et. al's algorithm [25]) $F_{AD}-F_{SA}$ $||x^*_{AD}-x^*_{SA}||$ 5 0.08 0.059 1.32e-13 9.59e-08 10 0.06 0.36 1.86e-09 1.08e-05 15 0.06 5.69 5.12e-09 1.66e-05 20 0.07 32.62 1.25e-09 6.09e-06 25 0.14 130.46 7.24e-10 3.75e-06 30 0.14 419.58 8.38e-09 1.72e-05
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