# American Institute of Mathematical Sciences

March  2021, 11(1): 45-61. doi: 10.3934/naco.2020014

## On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization

 1 School of Mathematical Sciences, Sunway University, Selangor, Malaysia 2 Department of Aerospace and Software Engineering, Gyeongsang National University, Jinju, Republic of Korea 3 School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Australia 4 Department of Electrical and Computer Engineering, Curtin University, Sarawak, Malaysia

* Corresponding author: M. S. LEE (Email: moksianglee@gmail.com)

Received  June 2019 Revised  August 2019 Published  February 2020

A bang-bang iteration method equipped with a component-wise line search strategy is introduced to solve unconstrained optimization problems. The main idea of this method is to formulate an unconstrained optimization problem as an optimal control problem to obtain an optimal trajectory. However, the optimal trajectory can only be generated by impulsive control variables and it is a straight line joining a guessed initial point to a minimum point. Thus, a priori bounds are imposed on the control variables in order to obtain a feasible solution. As a result, the optimal trajectory is made up of bang-bang control sub-arcs, which form an iterative model based on the Lyapunov function's theorem. This is to ensure monotonic decrease of the objective function value and convergence to a desirable minimum point. However, a chattering behavior may occur near the solution. To avoid this behavior, the Newton iterations are then applied to the proposed method via a two-phase approach to achieve fast convergence. Numerical experiments show that this new approach is efficient and cost-effective to solve the unconstrained optimization problems.

Citation: M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014
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##### References:
A backtracking procedure to obtain the next iterative point $x(k+1)$ from the current point $x(k)$ on an objective function with two variables
The AGD with rectangular search regions approaching the minimum point $x^{*}$ from the initial point $x(0)$
Performance profiles based on the CPU time
Performance profiles based on the number of iterations
The iterations of the AGDRN method on Booth function (40) in Phase-Ⅰ and Phase-Ⅱ
 Phase-Ⅰ $k$ $f(x)$ $\|g(x)\|$ $\|\Delta x\|$ 0 $3.79\times 10^{3}$ $3.69\times 10^{2}$ $1.80\times 10^{1}$ 1 $3.69\times 10^{2}$ $1.15\times 10^{2}$ $1.41\times 10^{1}$ 2 $3.69\times 10^{2}$ $1.15\times 10^{2}$ $1.27\times 10^{1}$ 3 $2.38\times 10^{1}$ $9.17\times 10^{1}$ $1.15\times 10^{1}$ 4 $4.04\times 10^{1}$ $3.62\times 10^{1}$ $3.09\times 10^{0}$ 5 $1.01\times 10^{1}$ $1.48\times 10^{1}$ $2.78\times 10^{0}$ 6 $4.51\times 10^{0}$ $4.29\times 10^{0}$ $7.52\times 10^{-1}$ 7 $2.10\times 10^{0}$ $2.95\times 10^{0}$ $6.76\times 10^{-1}$ 8 $1.60\times 10^{0}$ $2.60\times 10^{0}$ $1.83\times 10^{-1}$ 9 $1.22\times 10^{0}$ $2.28\times 10^{0}$ $1.64\times 10^{-1}$ 10 $9.13\times 10^{-1}$ $1.99\times 10^{0}$ $1.48\times 10^{-1}$ 11 $6.77\times 10^{-1}$ $1.74\times 10^{0}$ $1.33\times 10^{-1}$ 12 $4.96\times 10^{-1}$ $1.52\times 10^{0}$ $1.20\times 10^{-1}$ 13 $3.57\times 10^{-1}$ $1.33\times 10^{0}$ $1.08\times 10^{-1}$ 14 $2.52\times 10^{-1}$ $1.16\times 10^{0}$ $9.71\times 10^{-2}$ Phase-Ⅱ $j$ $f(x)$ $\|g(x)\|$ {det H} 0 $2.52\times 10^{-1}$ $1.16\times 10^{0}$ $3.60\times 10^{1}$ 1 $4.54\times 10^{-18}$ $4.34\times 10^{-9}$ $3.60\times 10^{1}$
 Phase-Ⅰ $k$ $f(x)$ $\|g(x)\|$ $\|\Delta x\|$ 0 $3.79\times 10^{3}$ $3.69\times 10^{2}$ $1.80\times 10^{1}$ 1 $3.69\times 10^{2}$ $1.15\times 10^{2}$ $1.41\times 10^{1}$ 2 $3.69\times 10^{2}$ $1.15\times 10^{2}$ $1.27\times 10^{1}$ 3 $2.38\times 10^{1}$ $9.17\times 10^{1}$ $1.15\times 10^{1}$ 4 $4.04\times 10^{1}$ $3.62\times 10^{1}$ $3.09\times 10^{0}$ 5 $1.01\times 10^{1}$ $1.48\times 10^{1}$ $2.78\times 10^{0}$ 6 $4.51\times 10^{0}$ $4.29\times 10^{0}$ $7.52\times 10^{-1}$ 7 $2.10\times 10^{0}$ $2.95\times 10^{0}$ $6.76\times 10^{-1}$ 8 $1.60\times 10^{0}$ $2.60\times 10^{0}$ $1.83\times 10^{-1}$ 9 $1.22\times 10^{0}$ $2.28\times 10^{0}$ $1.64\times 10^{-1}$ 10 $9.13\times 10^{-1}$ $1.99\times 10^{0}$ $1.48\times 10^{-1}$ 11 $6.77\times 10^{-1}$ $1.74\times 10^{0}$ $1.33\times 10^{-1}$ 12 $4.96\times 10^{-1}$ $1.52\times 10^{0}$ $1.20\times 10^{-1}$ 13 $3.57\times 10^{-1}$ $1.33\times 10^{0}$ $1.08\times 10^{-1}$ 14 $2.52\times 10^{-1}$ $1.16\times 10^{0}$ $9.71\times 10^{-2}$ Phase-Ⅱ $j$ $f(x)$ $\|g(x)\|$ {det H} 0 $2.52\times 10^{-1}$ $1.16\times 10^{0}$ $3.60\times 10^{1}$ 1 $4.54\times 10^{-18}$ $4.34\times 10^{-9}$ $3.60\times 10^{1}$
The iterations of the AGDRN method on Powell function (41) in Phase-Ⅰ and Phase-Ⅱ
 Phase-Ⅰ $k$ $f(x)$ $\|g(x)\|$ $\|\Delta x\|$ 0 $2.60\times 10^{2}$ $2.53\times 10^{2}$ $5.00\times 10^{0}$ 1 $6.56\times 10^{0}$ $1.31\times 10^{1}$ $3.54\times 10^{0}$ 2 $1.31\times 10^{0}$ $4.51\times 10^{0}$ $2.61\times 10^{0}$ 3 $-2.46\times 10^{-1}$ $9.70\times 10^{-1}$ $1.06\times 10^{0}$ Phase-Ⅱ $j$ $f(x)$ $\|g(x)\|$ {det H} 0 $-2.46\times 10^{-1}$ $9.70\times 10^{-1}$ $5.00\times 10^{-1}$ 1 $-5.49\times 10^{-1}$ $6.63\times 10^{-1}$ $1.36\times 10^{1}$ 2 $-5.82\times 10^{-1}$ $5.20\times 10^{-2}$ $1.09\times 10^{1}$ 3 $-5.82\times 10^{-1}$ $7.61\times 10^{-4}$ $1.06\times 10^{1}$ 4 $-5.82\times 10^{-1}$ $1.71\times 10^{-7}$ $1.06\times 10^{1}$
 Phase-Ⅰ $k$ $f(x)$ $\|g(x)\|$ $\|\Delta x\|$ 0 $2.60\times 10^{2}$ $2.53\times 10^{2}$ $5.00\times 10^{0}$ 1 $6.56\times 10^{0}$ $1.31\times 10^{1}$ $3.54\times 10^{0}$ 2 $1.31\times 10^{0}$ $4.51\times 10^{0}$ $2.61\times 10^{0}$ 3 $-2.46\times 10^{-1}$ $9.70\times 10^{-1}$ $1.06\times 10^{0}$ Phase-Ⅱ $j$ $f(x)$ $\|g(x)\|$ {det H} 0 $-2.46\times 10^{-1}$ $9.70\times 10^{-1}$ $5.00\times 10^{-1}$ 1 $-5.49\times 10^{-1}$ $6.63\times 10^{-1}$ $1.36\times 10^{1}$ 2 $-5.82\times 10^{-1}$ $5.20\times 10^{-2}$ $1.09\times 10^{1}$ 3 $-5.82\times 10^{-1}$ $7.61\times 10^{-4}$ $1.06\times 10^{1}$ 4 $-5.82\times 10^{-1}$ $1.71\times 10^{-7}$ $1.06\times 10^{1}$
Comparison of Phase-Ⅱ replacement with other methods to minimize the Gulf research and development function
 Methods $\boldsymbol{f(x^{*})}$ $\boldsymbol{\|g(x)\|}$ k CPU Time (s) AGDS $5.40\times 10^{-5}$ $6.40\times 10^{-9}$ 37 $9.06\times 10^{-1}$ BTR $5.40\times 10^{-5}$ $5.96\times 10^{-8}$ 81 $1.75\times 10^{0}$ AGDRN $3.11\times 10^{-2}$ $6.61\times 10^{-2}$ 500 $2.15\times 10^{1}$ AGD-RS $5.40\times 10^{-5}$ $1.35\times 10^{-7}$ 83 $1.63\times 10^{0}$
 Methods $\boldsymbol{f(x^{*})}$ $\boldsymbol{\|g(x)\|}$ k CPU Time (s) AGDS $5.40\times 10^{-5}$ $6.40\times 10^{-9}$ 37 $9.06\times 10^{-1}$ BTR $5.40\times 10^{-5}$ $5.96\times 10^{-8}$ 81 $1.75\times 10^{0}$ AGDRN $3.11\times 10^{-2}$ $6.61\times 10^{-2}$ 500 $2.15\times 10^{1}$ AGD-RS $5.40\times 10^{-5}$ $1.35\times 10^{-7}$ 83 $1.63\times 10^{0}$
Comparison of Phase-Ⅱ replacement with other methods to minimize the Kowalik Osborne function
 Methods $\boldsymbol{f(x^{*})}$ $\boldsymbol{\|g(x)\|}$ $\boldsymbol{k}$ CPU Time (s) AGDS $3.08\times 10^{-4}$ $8.60\times 10^{-8}$ 14 $6.09\times 10^{-1}$ BTR $3.08\times 10^{-4}$ $5.76\times 10^{-8}$ 16 $5.94\times 10^{-1}$ AGDRN $5.08\times 10^{-4}$ $3.84\times 10^{-3}$ 499 $2.38\times 10^{1}$ AGD-RS $3.08\times 10^{-4}$ $8.60\times 10^{-8}$ 14 $4.06\times 10^{-1}$
 Methods $\boldsymbol{f(x^{*})}$ $\boldsymbol{\|g(x)\|}$ $\boldsymbol{k}$ CPU Time (s) AGDS $3.08\times 10^{-4}$ $8.60\times 10^{-8}$ 14 $6.09\times 10^{-1}$ BTR $3.08\times 10^{-4}$ $5.76\times 10^{-8}$ 16 $5.94\times 10^{-1}$ AGDRN $5.08\times 10^{-4}$ $3.84\times 10^{-3}$ 499 $2.38\times 10^{1}$ AGD-RS $3.08\times 10^{-4}$ $8.60\times 10^{-8}$ 14 $4.06\times 10^{-1}$
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