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
References:
[1]

N. Andrei, An unconstrained optimization test functions collection, Adv. Model. Optim., 10 (2008), 147-161.   Google Scholar

[2]

D. Bushaw, Optimal discontinuous forcing terms, in Contributions to the Theory of Nonlinear Oscillations, Vol. IV (ed. S. Lefshetz), Princeton Univ. Press, Princeton, New Jersey, (1958), 29–52.  Google Scholar

[3]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

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R. Fletcher, Practical Methods of Optimization Volume 1: Unconstrained Optimization, John Wiley & Sons, 1987.  Google Scholar

[5]

I. Flüotz and H. Junr, Optimum and quasi-optimum control of third-and fourth-order systems, Technical Report 2, 1963. Google Scholar

[6]

A. Fuller, Relay control systems optimized for various performance criteria, in Automatic and Remote Control, Proc. First World Congress IFAC Moscow, vol. 1, 1960, 510–519. Google Scholar

[7]

P. E. GillW. MurrayM. A. Saunders and M. H. Wright, Computing forward-difference intervals for numerical optimization, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 310-321.  doi: 10.1137/0904025.  Google Scholar

[8]

B. S. Goh, Algorithms for unconstrained optimization problems via control theory, Journal of Optimization Theory and Applications, 92 (1997), 581-604.  doi: 10.1023/A:1022607507153.  Google Scholar

[9]

B. S. Goh, Greatest descent algorithms in unconstrained optimization, Journal of Optimization Theory and Applications, 142 (2009), 275-289.  doi: 10.1007/s10957-009-9533-4.  Google Scholar

[10]

B. S. Goh, Convergence of algorithms in optimization and solutions of nonlinear equations, Journal of Optimization Theory and Applications, 144 (2010), 43-55.  doi: 10.1007/s10957-009-9583-7.  Google Scholar

[11]

B. S. Goh, Approximate greatest descent methods for optimization with equality constraints, Journal of Optimization Theory and Applications, 148 (2011), 505-527.  doi: 10.1007/s10957-010-9765-3.  Google Scholar

[12]

B. S. Goh, W. J. Leong and K. L. Teo, Robustness of convergence proofs in numerical methods in unconstrained optimization, in Optimization and Control Methods in Industrial Engineering and Construction, Intelligent Systems, Control and Automation: Science and Engineering 72, Springer Netherlands, (2014), 1–9. Google Scholar

[13]

B. S. Goh and D. B. McDonald, Newton methods to solve a system of nonlinear algebraic equations, Journal of Optimization Theory and Applications, 164 (2015), 261-276.  doi: 10.1007/s10957-014-0544-4.  Google Scholar

[14]

A. Hedar, Studies on Metaheuristics for Continuous Global Optimization Problems, PhD Thesis, Kyoto University, Japan, 2004. Google Scholar

[15]

J. Kowalik and J. Morrison, Analysis of kinetic data for allosteric enzyme reactions as a nonlinear regression problem, Mathematical Biosciences, 2 (1968), 57-66.   Google Scholar

[16]

I. Kupka, The ubiquity of fuller's phenomenon, Nonlinear Controllability and Optimal Control, 133 (1990), 313-350.   Google Scholar

[17]

J. P. LaSalle, Time optimal control systems, Proceedings of the National Academy of Sciences of the United States of America. doi: 10.1073/pnas.45.4.573.  Google Scholar

[18]

J. P. LaSalle, Recent advances in liapunov stability theory, SIAM Review, 6 (1964), 1-11.  doi: 10.1137/1006001.  Google Scholar

[19]

M. S. LeeB. S. GohH. G. Harno and K. H. Lim, On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints, Numerical Algebra, Control & Optimization, 8 (2018), 325-336.  doi: 10.3934/naco.2018020.  Google Scholar

[20] G. Leitmann, The Calculus of Variations and Optimal Control, Vol. 20 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, 1981.   Google Scholar
[21]

J. J. MoréB. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software, ACM Transactions on Mathematical Software (TOMS), 7 (1981), 17-41.  doi: 10.1145/355934.355936.  Google Scholar

[22]

J. M. Ortega, Stability of difference equations and convergence of iterative processes, SIAM Journal on Numerical Analysis, 10 (1973), 268-282.  doi: 10.1137/0710026.  Google Scholar

[23] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, NY, 1970.   Google Scholar
[24]

B. T. Polyak, Newton's method and its use in optimization, European Journal of Operational Research, 181 (2007), 1086-1096.  doi: 10.1016/j.ejor.2005.06.076.  Google Scholar

[25]

M. I. Zelikin and V. F. Borisov, Regimes with increasingly more frequent switchings in optimal control problems, Tr. Mat. Inst. Akad. Nauk SSSR, 197 (1991), 85-167.   Google Scholar

[26]

M. I. Zelikin and V. F. Borisov, Theory of Chattering Control: With Applications to Astronautics, Robotics, Economics, and Engineering, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4612-2702-1.  Google Scholar

show all references

References:
[1]

N. Andrei, An unconstrained optimization test functions collection, Adv. Model. Optim., 10 (2008), 147-161.   Google Scholar

[2]

D. Bushaw, Optimal discontinuous forcing terms, in Contributions to the Theory of Nonlinear Oscillations, Vol. IV (ed. S. Lefshetz), Princeton Univ. Press, Princeton, New Jersey, (1958), 29–52.  Google Scholar

[3]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[4]

R. Fletcher, Practical Methods of Optimization Volume 1: Unconstrained Optimization, John Wiley & Sons, 1987.  Google Scholar

[5]

I. Flüotz and H. Junr, Optimum and quasi-optimum control of third-and fourth-order systems, Technical Report 2, 1963. Google Scholar

[6]

A. Fuller, Relay control systems optimized for various performance criteria, in Automatic and Remote Control, Proc. First World Congress IFAC Moscow, vol. 1, 1960, 510–519. Google Scholar

[7]

P. E. GillW. MurrayM. A. Saunders and M. H. Wright, Computing forward-difference intervals for numerical optimization, SIAM Journal on Scientific and Statistical Computing, 4 (1983), 310-321.  doi: 10.1137/0904025.  Google Scholar

[8]

B. S. Goh, Algorithms for unconstrained optimization problems via control theory, Journal of Optimization Theory and Applications, 92 (1997), 581-604.  doi: 10.1023/A:1022607507153.  Google Scholar

[9]

B. S. Goh, Greatest descent algorithms in unconstrained optimization, Journal of Optimization Theory and Applications, 142 (2009), 275-289.  doi: 10.1007/s10957-009-9533-4.  Google Scholar

[10]

B. S. Goh, Convergence of algorithms in optimization and solutions of nonlinear equations, Journal of Optimization Theory and Applications, 144 (2010), 43-55.  doi: 10.1007/s10957-009-9583-7.  Google Scholar

[11]

B. S. Goh, Approximate greatest descent methods for optimization with equality constraints, Journal of Optimization Theory and Applications, 148 (2011), 505-527.  doi: 10.1007/s10957-010-9765-3.  Google Scholar

[12]

B. S. Goh, W. J. Leong and K. L. Teo, Robustness of convergence proofs in numerical methods in unconstrained optimization, in Optimization and Control Methods in Industrial Engineering and Construction, Intelligent Systems, Control and Automation: Science and Engineering 72, Springer Netherlands, (2014), 1–9. Google Scholar

[13]

B. S. Goh and D. B. McDonald, Newton methods to solve a system of nonlinear algebraic equations, Journal of Optimization Theory and Applications, 164 (2015), 261-276.  doi: 10.1007/s10957-014-0544-4.  Google Scholar

[14]

A. Hedar, Studies on Metaheuristics for Continuous Global Optimization Problems, PhD Thesis, Kyoto University, Japan, 2004. Google Scholar

[15]

J. Kowalik and J. Morrison, Analysis of kinetic data for allosteric enzyme reactions as a nonlinear regression problem, Mathematical Biosciences, 2 (1968), 57-66.   Google Scholar

[16]

I. Kupka, The ubiquity of fuller's phenomenon, Nonlinear Controllability and Optimal Control, 133 (1990), 313-350.   Google Scholar

[17]

J. P. LaSalle, Time optimal control systems, Proceedings of the National Academy of Sciences of the United States of America. doi: 10.1073/pnas.45.4.573.  Google Scholar

[18]

J. P. LaSalle, Recent advances in liapunov stability theory, SIAM Review, 6 (1964), 1-11.  doi: 10.1137/1006001.  Google Scholar

[19]

M. S. LeeB. S. GohH. G. Harno and K. H. Lim, On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints, Numerical Algebra, Control & Optimization, 8 (2018), 325-336.  doi: 10.3934/naco.2018020.  Google Scholar

[20] G. Leitmann, The Calculus of Variations and Optimal Control, Vol. 20 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, 1981.   Google Scholar
[21]

J. J. MoréB. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software, ACM Transactions on Mathematical Software (TOMS), 7 (1981), 17-41.  doi: 10.1145/355934.355936.  Google Scholar

[22]

J. M. Ortega, Stability of difference equations and convergence of iterative processes, SIAM Journal on Numerical Analysis, 10 (1973), 268-282.  doi: 10.1137/0710026.  Google Scholar

[23] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, NY, 1970.   Google Scholar
[24]

B. T. Polyak, Newton's method and its use in optimization, European Journal of Operational Research, 181 (2007), 1086-1096.  doi: 10.1016/j.ejor.2005.06.076.  Google Scholar

[25]

M. I. Zelikin and V. F. Borisov, Regimes with increasingly more frequent switchings in optimal control problems, Tr. Mat. Inst. Akad. Nauk SSSR, 197 (1991), 85-167.   Google Scholar

[26]

M. I. Zelikin and V. F. Borisov, Theory of Chattering Control: With Applications to Astronautics, Robotics, Economics, and Engineering, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4612-2702-1.  Google Scholar

Figure 1.  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
Figure 2.  The AGD with rectangular search regions approaching the minimum point $ x^{*} $ from the initial point $ x(0) $
Figure 3.  Performance profiles based on the CPU time
Figure 4.  Performance profiles based on the number of iterations
Table5 
Table6 
Table 1.  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} $
Table 2.  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} $
Table 3.  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} $
Table 4.  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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