Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions
Abstract
:1. Introduction
Paper Contributions and Structure
 The review of techniques proposed to tackle linearly constrained problems within the framework of DIRECTtype algorithms.
 Introduction of a novel and distinctive DIRECTtype algorithm explicitly designed for nonconvex problems involving linear constraints.
 The substantial enhancement of the DIRECTGOLib v1.3 benchmark library by incorporating 34 linearyconstrained test problems.
 The provision of the novel algorithm developed as an opensource resource to ensure the full reproducibility and reusability of all results.
2. Materials and Methods
2.1. The Original DIRECT Algorithm for BoxConstrained Global Optimization
2.2. Extensions of the DIRECT Algorithm for Problems with Constraints
2.2.1. Approaches Based on Simplicial Partitioning
2.2.2. Penalty and Auxiliary Function Approaches
2.2.3. Filtering Approach
2.2.4. Alternative Approaches without Utilizing Constraint Information
3. Description of the Proposed mBIRECTvGL Algorithm
3.1. Efficient Bijective Mapping: Construction and Methodological Details
3.2. Integrating Mapping Techniques in DIRECTBased Framework
Selection of the Most Promising Regions Using a TwoStepBased Approach
3.3. Description of a Novel Algorithm (mBIRECTvGL)
Algorithm 1 The mBIRECTvGL algorithm 

3.4. Convergence Properties of the mBIRECTvGL Algorithm
4. Results and Discussions
4.1. Foundation of Solver Comparisons and Design of Experimental Setup
4.2. Analysis of the Overall Performance of Algorithms
4.3. Statistical Analysis of the Results
5. Conclusions and Future Prospects
Author Contributions
Funding
Data Availability Statement
 https://github.com/blockchaingroup/DIRECTGOLib (accessed on 15 June 2023),
 https://zenodo.org/record/8046086 (accessed on 16 June 2023).
 https://github.com/blockchaingroup/DIRECTGO (accessed on 15 June 2023).
Conflicts of Interest
Appendix A. Linearly Constrained Test Problems from the DIRECTGOLib v1.3 Benchmark Library
#  Name  Ref.  n  Con.  AC  Variable Bounds $\left(\mathit{D}\right)$  Optimum $\left({\mathit{f}}^{*}\right)$ 

1  avgasa  [66]  8  10  3  ${[0,1]}^{n}$  $3.4201$ 
2  avgasb  [66]  8  10  4  ${[0,1]}^{n}$  $4.4832$ 
3  biggsc4  [66]  4  13  3  ${[0,5]}^{n}$  $24.5000$ 
4  Bunnag1  [66]  3  1  1  ${[0,3]}^{n}$  $0.1111$ 
5  Bunnag2  [66]  4  2  1  ${[0,4]}^{n}$  $6.4052$ 
6  Bunnag3  [66]  5  3  1  $[0,3]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{[0,4]}^{2}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2]$  $16.3693$ 
7  Bunnag4  [66]  6  2  1  ${[0,1]}^{5}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,20]$  $213.0470$ 
8  Bunnag5  [66]  6  5  1  $[0,2]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,8]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{[0,1]}^{2}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2]$  $11.0000$ 
9  Bunnag6  [66]  10  11  3  ${[0,1]}^{n}$  $268.0146$ 
10  Bunnag7  [66]  10  5  0  ${[0,1]}^{n}$  $39.0000$ 
11  Bunnag8  [66]  5  1  0  ${[0,1]}^{n}$  $17.0000$ 
12  Bunnag10  [66]  20  10  5  ${[0,100]}^{n}$  $394.7506$ 
13  Bunnag11  [66]  20  10  5  ${[0,100]}^{n}$  $884.7506$ 
14  Bunnag12  [66]  20  10  5  ${[0,100]}^{n}$  $8695.0122$ 
15  Bunnag13  [66]  20  10  0  ${[0,100]}^{n}$  $754.7506$ 
16  Bunnag14  [66]  20  10  0  ${[0,100]}^{n}$  $4118.725$ 
17  Bunnag15  [66]  20  10  2  ${[0,100]}^{n}$  $\mathrm{49,318.0180}$ 
18  ex2_1_1  [66]  5  1  1  ${[0,20]}^{n}$  $4525.0000$ 
19  ex2_1_2  [66]  6  2  1  ${[0,1]}^{5}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,100]$  $213.0000$ 
20  expfita  [66]  5  2  0  ${[0,20]}^{n}$  $0.0342$ 
21  expfitb  [66]  5  2  0  ${[0,20]}^{n}$  $0.0860$ 
22  expfitc  [66]  5  2  0  ${[0,20]}^{n}$  $0.3567$ 
23  G01  [66]  13  9  6  ${[0,10]}^{9}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{[0,100]}^{3}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,10]$  $15.0000$ 
24  Genocop7  [66]  6  2  1  ${[0,1]}^{5}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,100]$  $413.0000$ 
25  Genocop9  [66]  3  5  2  ${[0,3]}^{n}$  $2.4714$ 
26  Genocop10  [66]  4  2  1  $[0,3]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{[0,10]}^{2}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,1]$  $4.5284$ 
27  Horst1  [67]  2  3  1  $[0,3]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2]$  $1.0625$ 
28  Horst2  [67]  2  3  2  $[0,2.5]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2]$  $6.8995$ 
29  Horst3  [67]  2  3  0  $[0,1]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,1.5]$  $0.4444$ 
30  Horst4  [67]  3  4  2  $[0.5,2]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,3]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2.8]$  $6.0858$ 
31  Horst5  [67]  3  4  2  $[0,1.2]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,1.2]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,1.7]$  $3.7220$ 
32  Horst6  [67]  3  7  2  $[0,6]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,5.0279]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2.6]$  $32.5793$ 
33  Horst7  [67]  3  4  2  $[0,6]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{[0,3]}^{2}$  $52.8774$ 
34  hs021  [66]  2  1  0  $[2,50]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[50,10]$  $99.9600$ 
35  hs021mod  [66]  7  1  1  $[2,50]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[50,50]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,50]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[2,10]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[10,10]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[10,0]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,10]$  $4.0400$ 
36  hs024  [66]  2  3  2  ${[0,5]}^{n}$  $1.0000$ 
37  hs036  [66]  3  1  1  $[0,20]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,11]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,15]$  $3300.0000$ 
38  hs037  [66]  3  2  1  ${[0,42]}^{n}$  $3456.0000$ 
39  hs038  [66]  4  2  0  ${[10,10]}^{n}$  $0.0000$ 
40  hs044  [66]  4  6  2  ${[0,42]}^{n}$  $15.0000$ 
41  hs076  [66]  4  3  1  $[0,1]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,3]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{[0,1]}^{2}$  $4.6818$ 
42  hs086  [66]  5  1  0  ${[0,10]}^{n}$  $351.7236$ 
43  hs118  [66]  15  17  9  ${[0,100]}^{n}$  $553.9246$ 
44  hs268  [66]  5  5  2  ${[0,10]}^{n}$  $\mathrm{63,126.1111}$ 
45  Ji1  [66]  3  4  1  ${[0,10]}^{n}$  $4.0907$ 
46  Ji2  [66]  3  2  0  ${[0,10]}^{n}$  $3.0029$ 
47  Ji3  [66]  2  1  0  ${[0,10]}^{n}$  $4.6758$ 
48  ksip  [66]  10  20  2  ${[0,10]}^{n}$  $12.1448$ 
49  Michalewicz1  [66]  2  3  0  ${[0,10]}^{n}$  $1.0000$ 
50  P9  [66]  3  9  2  $[{10}^{5},3]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{[{10}^{5},4]}^{2}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{[0,2]}^{2}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,6]$  $13.4019$ 
51  P14  [66]  3  4  2  $[{10}^{5},3]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[{10}^{5},4]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,2]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,1]$  $4.5142$ 
52  s224  [66]  2  4  1  $[0,6]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,11]$  $304.0000$ 
53  s231  [66]  2  2  0  ${[10,10]}^{n}$  $0.0000$ 
54  s232  [66]  2  3  2  ${[0,100]}^{n}$  $1.0000$ 
55  s250  [66]  3  2  1  $[0,20]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,11]\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}[0,42]$  $3300.0000$ 
56  s251  [66]  3  1  1  ${[0,42]}^{n}$  $3456.0000$ 
57  s253  [66]  3  1  0  ${[0,100]}^{n}$  $120.0000$ 
58  s268  [66]  5  5  2  ${[0,2]}^{n}$  $7.0913$ 
59  s277  [66]  4  4  4  ${[0,10]}^{n}$  $5.0762$ 
60  s278  [66]  6  6  6  ${[0,10]}^{n}$  $7.8385$ 
61  s279  [66]  8  8  8  ${[0,10]}^{n}$  $10.6060$ 
62  s280  [66]  10  10  10  ${[0,10]}^{n}$  $13.3754$ 
63  s331  [66]  2  1  0  ${[0.0001,10]}^{n}$  $4.2584$ 
64  s340  [66]  3  1  1  ${[0.0001,10]}^{n}$  $0.0540$ 
65  s354  [66]  4  1  1  ${[0,20]}^{n}$  $0.2596$ 
66  s359  [66]  5  14  4  ${[0,10]}^{n}$  $\mathrm{563,335.5491}$ 
67  zecevic2  [66]  2  2  1  ${[0,10]}^{n}$  $4.1250$ 
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Average Number of Function Evaluations  

Algorithm  Fails  Overall  $\mathit{n}\le 5$  $\mathit{n}\ge 6$  AC ^{1}  NAC ^{2} 
mBIRECTvGL  $9/67$  $\mathrm{155,031}$  $\mathrm{70,140}$  $\mathrm{340,980}$  $\mathrm{107,449}$  $\mathrm{294,975}$ 
LcDISIMPLv  $18/67$  $\mathrm{269,349}$  $\mathrm{66,192}$  $\mathrm{714,356}$  $\mathrm{260,327}$  $\mathrm{295,882}$ 
LcDISIMPLc  $22/67$  $\mathrm{333,860}$  $\mathrm{137,852}$  $\mathrm{763,208}$  $\mathrm{396,432}$  $\mathrm{295,026}$ 
glcSolve  $23/67$  $\mathrm{370,703}$  $\mathrm{134,031}$  $\mathrm{889,126}$  $\mathrm{166,726}$  $\mathrm{572,827}$ 
DIRECTGLc  $16/67$  $\mathrm{286,976}$  $\mathrm{114,737}$  $\mathrm{664,259}$  $\mathrm{281,871}$  $\mathrm{301,988}$ 
DIRECTGLce  $14/67$  $\mathrm{283,836}$  $\mathrm{121,677}$  $\mathrm{639,040}$  $\mathrm{277,607}$  $\mathrm{302,154}$ 
Algorithm  ${10}^{2}$  ${10}^{3}$  ${10}^{4}$  ${10}^{5}$  ${10}^{6}$ 

LcDISIMPLv  $4.3758\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$  $4.1523\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$  $1.0205\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$  $2.9316\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$  $2.1367\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ 
LcDISIMPLc  $7.2762\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$  $2.8887\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$  $1.8188\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$  $6.3294\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$  $8.0517\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$ 
glcSolve  $1.4166\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{7}$  $8.7745\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$  $4.8968\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$  $1.4354\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$  $1.1472\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ 
DIRECTGLc  $3.5571\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}$  $2.2431\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$  $2.6512\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$  $1.1858\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$  $1.4097\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ 
DIRECTGLce  $4.2110\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}$  $4.5098\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{10}$  $7.0552\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{7}$  $2.4254\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$  $3.9474\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{2}$ 
Algorithm  ${10}^{2}$  ${10}^{3}$  ${10}^{4}$  ${10}^{5}$  ${10}^{6}$ 

mBIRECTvGL  $1.8060$  $2.2015$  $2.6866$  $2.8806$  $2.9030$ 
LcDISIMPLv  $2.4478$  $2.9179$  $3.3060$  $3.7164$  $3.7910$ 
LcDISIMPLc  $4.0373$  $3.5373$  $3.7537$  $3.9552$  $3.9701$ 
glcSolve  $3.3134$  $3.1866$  $3.4851$  $3.5672$  $3.8059$ 
DIRECTGLc  $4.5672$  $4.1866$  $3.5224$  $3.3731$  $3.2761$ 
DIRECTGLce  $4.8284$  $4.9701$  $4.2463$  $3.5075$  $3.1567$ 
pvalue  $8.4848\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{32}$  $2.5717\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{23}$  $3.1391\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$  $7.3195\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$  $3.3810\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$ 
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Stripinis, L.; Paulavičius, R. Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions. Mathematics 2023, 11, 2920. https://doi.org/10.3390/math11132920
Stripinis L, Paulavičius R. Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions. Mathematics. 2023; 11(13):2920. https://doi.org/10.3390/math11132920
Chicago/Turabian StyleStripinis, Linas, and Remigijus Paulavičius. 2023. "Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions" Mathematics 11, no. 13: 2920. https://doi.org/10.3390/math11132920