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## Optimizing over Pareto set of semistrictly quasiconcave vector maximization and application to stochastic portfolio selection

 School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No. 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam

*Corresponding author: Tran Ngoc Thang

Received  July 2021 Revised  January 2022 Early access March 2022

Optimization over Pareto set of a semistrictly quasiconcave vector maximization problem has many applications in economics and technology but it is a challenging task because of the nonconvexity of objective functions and constraint sets. In this article, we propose a novel approach, which is a Branch-and-Bound algorithm for maximizing a composite function $\varphi(f(x))$ over the non-dominated solution set of the $p$-objective programming problem, where $p\geq 2, p \in \mathbb{N},$ the function $\varphi$ is increasing and the objective function $f$ is semistrictly quasiconcave. By utilizing the nice properties of Pareto set to define the partitions of branch and bound scheme, the proposed algorithms are promised to be more accurate and efficient than ones using the multi-objective evolutionary approach such as NSGA-III. This is validated by some computational experiments. The Stochastic Portfolio Selection Problem is chosen as an application of our algorithm, where Sharpe ratio is a semistrictly quasiconcave objective function.

Citation: Nguyen Duc Vuong, Tran Ngoc Thang. Optimizing over Pareto set of semistrictly quasiconcave vector maximization and application to stochastic portfolio selection. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022029
##### References:
 [1] M. Avriel, W. Diewert, E. Walter, S. Schaible and Z. Isreal, Generalized Concavity, Society for Industrial and Applied Mathematics, 2010. doi: 10.1137/1.9780898719437.ch1. [2] H. P. Benson, An Outcome Space Branch and Bound-Outer Approximation Algorithm for Convex Multiplicative Programming, Journal of Global Optimization, 15 (1999), 315-342.  doi: 10.1023/A:1008316429329. [3] H. P. Benson, An outcome space algorithm for optimization over the weakly efficient set of a multiple objective nonlinear programming problem, Journal of Global Optimization, 52 (2012), 553-574.  doi: 10.1007/s10898-011-9786-y. [4] H. P. Benson, Outcome-Space Cutting-Plane Algorithm for Linear Multiplicative Programming, Journal of Optimization Theory and Applications, 104 (2000), 301-322.  doi: 10.1023/A:1004657629105. [5] H. P. Benson and D. Lee, Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem, Journal of Optimization Theory and Applications, 88 (1996), 77-105.  doi: 10.1007/BF02192023. [6] H. Bonnel and J. Collonge, Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case, Journal of Global Optimization, 62 (2014), 481-505.  doi: 10.1007/s10898-014-0257-0. [7] J. P. Dauer and T. A. Fosnaugh, Optimization over the efficient set, Journal of Global Optimization, 7 (1995), 261-277.  doi: 10.1007/BF01279451. [8] J. Fülöp and L. D. Muu, Branch-and-Bound Variant of an Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem, Journal of Optimization Theory and Applications, 105 (2000), 37-54.  doi: 10.1023/A:1004657827134. [9] A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers and Industrial Engineering, 137 (2019), 106090. [10] A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm, Numerical Algebra, Control and Optimization, 9 (2019), 187-209.  doi: 10.3934/naco.2019014. [11] N. Gülpinar and E. Çanakoḡlu, Robust portfolio selection problem under temperature uncertainty, European Journal of Operational Research, 256 (2017), 500-523.  doi: 10.1016/j.ejor.2016.05.046. [12] A. Hajnoori, M. Amiri and A. Alimi, Forecasting stock price using grey-fuzzy technique and portfolio optimization by invasive weed optimization algorithm, Decision Science Letters, 2 (2013), 175-184. [13] B. Jaumard, C. Meyer and H. Tuy, Generalized convex multiplicative programming via quasiconcave minimization, Journal of Global Optimization, 10 (1997), 229–256. doi: 10.1023/A:1008203116882. [14] D. Kalyanmoy and J. Himanshu, An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, IEEE Transactions on Evolutionary Computation, 18 (2014), 577-601. [15] N. T. B. Kim, An algorithm for optimizing over the non-dominated set, Vietnam J. Math, 28 (2000), 329-340. [16] N. T. B. Kim and L. D. Muu, On the projection of the non-dominated set and potential application, Optimization, 51 (2002), 401-421.  doi: 10.1080/02331930290019486. [17] N. T. B. Kim and T. N. Thang, Optimization over the efficient set of a bicriteria convex programming problem, Pacific Journal of Optimization, 9 (2013), 103-115. [18] Z. Liu and M. Ehrgott, Primal and dual algorithms for optimization over the efficient set, Optimization, 67 (2018), 1661-1686.  doi: 10.1080/02331934.2018.1484922. [19] F. Lolli, R. Gamberini, A. Regattieri, E. Balugani, T. Gatos and S. Gucci, Single-hidden layer neural networks for forecasting intermittent demand, International Journal of Production Economics, 183 (2017), 116-128. [20] K. Lu, S. Mizuno and J. Shi, A new mixed integer programming approach for optimization over the efficient set of a multiobjective linear programming problem, Optimization Letters, 14 (2020), 2323-2333.  doi: 10.1007/s11590-020-01554-7. [21] T. Matsui, NP-hardness of linear multiplicative programming and related problems, Journal of Global Optimization, 9 (1996), 113-119.  doi: 10.1007/BF00121658. [22] T. Munson, Mesh shape-quality optimization using the inverse mean-ratio metric, Math. Program., 110 (2007), 561-590.  doi: 10.1007/s10107-006-0014-3. [23] L. D. Muu, A convex-concave programming method for optimizing over the non-dominated set, Acta Math. Vietnam, 25 (2000), 67-85. [24] J. Philip, Algorithms for the vector maximization problem, Mathematical Programming, 3 (1972), 207-229.  doi: 10.1007/BF01584543. [25] H. X. Phu, On efficient sets in $\mathbb{R}^{2}$, Vietnam Journal of Mathematics, 33 (2005), 463-468. [26] M. R. Rao, Cluster analysis and mathematical programming, Journal of the American Statistical Association, 66 (1971), 622-626. [27] M. Sakawa, H. Yano and I. Nishizaki, Linear and Multiobjective Programming with Fuzzy Stochastic Extensions, vol. 203, Springer US, 2013. doi: 10.1007/978-1-4614-9399-0. [28] W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, The Journal of Finance, 19 (1964), 425–442. [29] T. N. Thang, S. Vijender, A. Dao, N. T. N. Anh and P. Hai, A monotonic optimization approach for solving strictly quasiconvex multiobjective programming problems, Journal of Intelligent and Fuzzy Systems, 38 (2020), 6053-6063. [30] N. V. Thoai, Decomposition branch and bound algorithm for optimization problems over efficient sets, Journal of Industrial and Management Optimization, 4 (2008), 647-660.  doi: 10.3934/jimo.2008.4.647. [31] N. V. Thoai, Reverse convex programming approach in the space of extreme criteria for optimization over efficient sets, Journal of Optimization Theory and Applications, 147 (2010), 263-277.  doi: 10.1007/s10957-010-9721-2. [32] H. Tuy and N. D. Nghia, Reverse polyblock approximation for generalized multiplicative/fractional programming, Vietnam Journal of Mathematics, 31 (2003), 391-402. [33] Y. Yamamoto, Optimization over the efficient set: overview, Journal of Global Optimization, 22 (2002), 285-317.  doi: 10.1023/A:1013875600711. [34] I. Yevseyeva, E. B. Lenselink, A. de Vries, A. P. IJzerman, A. H. Deutz and M. T. Emmerich, Application of portfolio optimization to drug discovery, Information Sciences, 475 (2019), 29-43.

show all references

##### References:
 [1] M. Avriel, W. Diewert, E. Walter, S. Schaible and Z. Isreal, Generalized Concavity, Society for Industrial and Applied Mathematics, 2010. doi: 10.1137/1.9780898719437.ch1. [2] H. P. Benson, An Outcome Space Branch and Bound-Outer Approximation Algorithm for Convex Multiplicative Programming, Journal of Global Optimization, 15 (1999), 315-342.  doi: 10.1023/A:1008316429329. [3] H. P. Benson, An outcome space algorithm for optimization over the weakly efficient set of a multiple objective nonlinear programming problem, Journal of Global Optimization, 52 (2012), 553-574.  doi: 10.1007/s10898-011-9786-y. [4] H. P. Benson, Outcome-Space Cutting-Plane Algorithm for Linear Multiplicative Programming, Journal of Optimization Theory and Applications, 104 (2000), 301-322.  doi: 10.1023/A:1004657629105. [5] H. P. Benson and D. Lee, Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem, Journal of Optimization Theory and Applications, 88 (1996), 77-105.  doi: 10.1007/BF02192023. [6] H. Bonnel and J. Collonge, Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case, Journal of Global Optimization, 62 (2014), 481-505.  doi: 10.1007/s10898-014-0257-0. [7] J. P. Dauer and T. A. Fosnaugh, Optimization over the efficient set, Journal of Global Optimization, 7 (1995), 261-277.  doi: 10.1007/BF01279451. [8] J. Fülöp and L. D. Muu, Branch-and-Bound Variant of an Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem, Journal of Optimization Theory and Applications, 105 (2000), 37-54.  doi: 10.1023/A:1004657827134. [9] A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers and Industrial Engineering, 137 (2019), 106090. [10] A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm, Numerical Algebra, Control and Optimization, 9 (2019), 187-209.  doi: 10.3934/naco.2019014. [11] N. Gülpinar and E. Çanakoḡlu, Robust portfolio selection problem under temperature uncertainty, European Journal of Operational Research, 256 (2017), 500-523.  doi: 10.1016/j.ejor.2016.05.046. [12] A. Hajnoori, M. Amiri and A. Alimi, Forecasting stock price using grey-fuzzy technique and portfolio optimization by invasive weed optimization algorithm, Decision Science Letters, 2 (2013), 175-184. [13] B. Jaumard, C. Meyer and H. Tuy, Generalized convex multiplicative programming via quasiconcave minimization, Journal of Global Optimization, 10 (1997), 229–256. doi: 10.1023/A:1008203116882. [14] D. Kalyanmoy and J. Himanshu, An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, IEEE Transactions on Evolutionary Computation, 18 (2014), 577-601. [15] N. T. B. Kim, An algorithm for optimizing over the non-dominated set, Vietnam J. Math, 28 (2000), 329-340. [16] N. T. B. Kim and L. D. Muu, On the projection of the non-dominated set and potential application, Optimization, 51 (2002), 401-421.  doi: 10.1080/02331930290019486. [17] N. T. B. Kim and T. N. Thang, Optimization over the efficient set of a bicriteria convex programming problem, Pacific Journal of Optimization, 9 (2013), 103-115. [18] Z. Liu and M. Ehrgott, Primal and dual algorithms for optimization over the efficient set, Optimization, 67 (2018), 1661-1686.  doi: 10.1080/02331934.2018.1484922. [19] F. Lolli, R. Gamberini, A. Regattieri, E. Balugani, T. Gatos and S. Gucci, Single-hidden layer neural networks for forecasting intermittent demand, International Journal of Production Economics, 183 (2017), 116-128. [20] K. Lu, S. Mizuno and J. Shi, A new mixed integer programming approach for optimization over the efficient set of a multiobjective linear programming problem, Optimization Letters, 14 (2020), 2323-2333.  doi: 10.1007/s11590-020-01554-7. [21] T. Matsui, NP-hardness of linear multiplicative programming and related problems, Journal of Global Optimization, 9 (1996), 113-119.  doi: 10.1007/BF00121658. [22] T. Munson, Mesh shape-quality optimization using the inverse mean-ratio metric, Math. Program., 110 (2007), 561-590.  doi: 10.1007/s10107-006-0014-3. [23] L. D. Muu, A convex-concave programming method for optimizing over the non-dominated set, Acta Math. Vietnam, 25 (2000), 67-85. [24] J. Philip, Algorithms for the vector maximization problem, Mathematical Programming, 3 (1972), 207-229.  doi: 10.1007/BF01584543. [25] H. X. Phu, On efficient sets in $\mathbb{R}^{2}$, Vietnam Journal of Mathematics, 33 (2005), 463-468. [26] M. R. Rao, Cluster analysis and mathematical programming, Journal of the American Statistical Association, 66 (1971), 622-626. [27] M. Sakawa, H. Yano and I. Nishizaki, Linear and Multiobjective Programming with Fuzzy Stochastic Extensions, vol. 203, Springer US, 2013. doi: 10.1007/978-1-4614-9399-0. [28] W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, The Journal of Finance, 19 (1964), 425–442. [29] T. N. Thang, S. Vijender, A. Dao, N. T. N. Anh and P. Hai, A monotonic optimization approach for solving strictly quasiconvex multiobjective programming problems, Journal of Intelligent and Fuzzy Systems, 38 (2020), 6053-6063. [30] N. V. Thoai, Decomposition branch and bound algorithm for optimization problems over efficient sets, Journal of Industrial and Management Optimization, 4 (2008), 647-660.  doi: 10.3934/jimo.2008.4.647. [31] N. V. Thoai, Reverse convex programming approach in the space of extreme criteria for optimization over efficient sets, Journal of Optimization Theory and Applications, 147 (2010), 263-277.  doi: 10.1007/s10957-010-9721-2. [32] H. Tuy and N. D. Nghia, Reverse polyblock approximation for generalized multiplicative/fractional programming, Vietnam Journal of Mathematics, 31 (2003), 391-402. [33] Y. Yamamoto, Optimization over the efficient set: overview, Journal of Global Optimization, 22 (2002), 285-317.  doi: 10.1023/A:1013875600711. [34] I. Yevseyeva, E. B. Lenselink, A. de Vries, A. P. IJzerman, A. H. Deutz and M. T. Emmerich, Application of portfolio optimization to drug discovery, Information Sciences, 475 (2019), 29-43.
Initial Partition
Second partitioning step
Computational result in the case $h(x) = \varphi_{0}\left(f(x)\right)$
 Iteration $y_{k}^{\text{best}}$ $y_{k}^{\text{new}}$ $U_{k}$ $L_{k}$ $\text{Gap}_{k}$ $k=1$ $(1.7620,7.7479)^{T}$ $(1.7620,7.7479)^{T}$ $13.6517$ $6.1382$ $7.5135$ $\ldots$ $k=13$ $(1.0256,9.5264)^{T}$ $(1.0183,9.5985)^{T}$ $9.7703$ $9.7406$ $0.0297$ $k=14$ $(1.0256,9.5264)^{T}$ $(1.0010,10.6920)^{T}$ $9.7703$ $9.7438$ $0.0266$
 Iteration $y_{k}^{\text{best}}$ $y_{k}^{\text{new}}$ $U_{k}$ $L_{k}$ $\text{Gap}_{k}$ $k=1$ $(1.7620,7.7479)^{T}$ $(1.7620,7.7479)^{T}$ $13.6517$ $6.1382$ $7.5135$ $\ldots$ $k=13$ $(1.0256,9.5264)^{T}$ $(1.0183,9.5985)^{T}$ $9.7703$ $9.7406$ $0.0297$ $k=14$ $(1.0256,9.5264)^{T}$ $(1.0010,10.6920)^{T}$ $9.7703$ $9.7438$ $0.0266$
Computational result in the remaining cases of $\varphi(y)$
 $\varphi(y)$ \#Iter $y^{\varepsilon}$ $\varphi_{\text{opt}}$ Time $y_{1}^{0.5}y_{2}^{3}$ $9$ $(14.3236,2.3804)^{T}$ $51.0477$ $1.6600$ $0.4y_{1}^{0.5}y_{2}^{3}+0.9y_{1}^{2}y_{2}^{0.25}$ $36$ $(6.7177,4.2559)^{T}$ $138.2533$ $5.1700$ $y_{1}^{0.25}/(10-y_{2}^{0.5})$ $3$ $(1.0129,9.7559)^{T}$ $0.1459$ $0.5000$ $0.3\log y_{1}+1.5\log y_{2}$ $7$ $(14.3236,2.3804)$ $2.0995$ $1.4700$
 $\varphi(y)$ \#Iter $y^{\varepsilon}$ $\varphi_{\text{opt}}$ Time $y_{1}^{0.5}y_{2}^{3}$ $9$ $(14.3236,2.3804)^{T}$ $51.0477$ $1.6600$ $0.4y_{1}^{0.5}y_{2}^{3}+0.9y_{1}^{2}y_{2}^{0.25}$ $36$ $(6.7177,4.2559)^{T}$ $138.2533$ $5.1700$ $y_{1}^{0.25}/(10-y_{2}^{0.5})$ $3$ $(1.0129,9.7559)^{T}$ $0.1459$ $0.5000$ $0.3\log y_{1}+1.5\log y_{2}$ $7$ $(14.3236,2.3804)$ $2.0995$ $1.4700$
Expected returns and covariance matrix of stocks
 Stock Expected Profit Covariance Matrix IBM BAC AAPL MSFT DGX IBM 0.400% 0.0065 0.0010 0.0024 0.0030 0.0024 BAC 1.236% 0.0010 0.0016 0.0013 0.0006 0.0004 AAPL 4.085% 0.0024 0.0013 0.0127 0.0010 0.0014 MSFT 0.513% 0.0030 0.0006 0.0010 0.0039 -0.0002 DGX 1.006% 0.0024 0.0004 0.0014 -0.0002 0.0056
 Stock Expected Profit Covariance Matrix IBM BAC AAPL MSFT DGX IBM 0.400% 0.0065 0.0010 0.0024 0.0030 0.0024 BAC 1.236% 0.0010 0.0016 0.0013 0.0006 0.0004 AAPL 4.085% 0.0024 0.0013 0.0127 0.0010 0.0014 MSFT 0.513% 0.0030 0.0006 0.0010 0.0039 -0.0002 DGX 1.006% 0.0024 0.0004 0.0014 -0.0002 0.0056
Optimal Portfolio associated with $\lambda$
 $\lambda$ $x^{{\text {best}}}$ $\phi$ $T$ 0.2 (0.0000, 0.0000, 0.3552, 0.0711, 0.5736) 0.3124 1.3500 0.5 (0.0013, 0.0013, 0.3939, 0.1120, 0.4919) 0.2405 1.1300 0.8 (0.0001, 0.0007, 0.9984, 0.0003, 0.0005) 0.1019 0.9000 (a) BB-2 Algorithm $\lambda$ $x^{{\text {best}}}$ $\phi$ $T$ 0.2 (0.0000, 0.0000, 0.3994, 0.0333, 0.5672) 0.3119 59.0200 0.5 (0.0000, 0.0000, 0.4075, 0.0350, 0.5575) 0.2038 59.1000 0.8 (0.0000, 0.0000, 0.9975, 0.0000, 0.0025) 0.1018 58.9000 (b) NSGA-III Algorithm
 $\lambda$ $x^{{\text {best}}}$ $\phi$ $T$ 0.2 (0.0000, 0.0000, 0.3552, 0.0711, 0.5736) 0.3124 1.3500 0.5 (0.0013, 0.0013, 0.3939, 0.1120, 0.4919) 0.2405 1.1300 0.8 (0.0001, 0.0007, 0.9984, 0.0003, 0.0005) 0.1019 0.9000 (a) BB-2 Algorithm $\lambda$ $x^{{\text {best}}}$ $\phi$ $T$ 0.2 (0.0000, 0.0000, 0.3994, 0.0333, 0.5672) 0.3119 59.0200 0.5 (0.0000, 0.0000, 0.4075, 0.0350, 0.5575) 0.2038 59.1000 0.8 (0.0000, 0.0000, 0.9975, 0.0000, 0.0025) 0.1018 58.9000 (b) NSGA-III Algorithm
Computational results of Example 6.5
 $n$ $m$ Iter $Ub$ $Lb$ Gap Time $60$ $40$ $5.2$ $4.8458$ $4.8312$ $0.0025$ $0.88$ $70$ $50$ $6.4$ $6.1537$ $6.1458$ $0.0011$ $1.31$ $80$ $80$ $5.8$ $2.3188$ $2.3135$ $0.0016$ $1.77$ $100$ $60$ $4.4$ $5.3968$ $5.3917$ $0.0008$ $2.45$ $100$ $80$ $5.1$ $1.5828$ $1.5707$ $0.0047$ $3.94$ $120$ $120$ $4.7$ $3.8471$ $3.8331$ $0.0029$ $6.12$ $150$ $100$ $4.9$ $6.6608$ $6.6340$ $6.6340$ $7.34$ $150$ $120$ $6.2$ $8.7834$ $8.7580$ $0.0026$ $9.21$
 $n$ $m$ Iter $Ub$ $Lb$ Gap Time $60$ $40$ $5.2$ $4.8458$ $4.8312$ $0.0025$ $0.88$ $70$ $50$ $6.4$ $6.1537$ $6.1458$ $0.0011$ $1.31$ $80$ $80$ $5.8$ $2.3188$ $2.3135$ $0.0016$ $1.77$ $100$ $60$ $4.4$ $5.3968$ $5.3917$ $0.0008$ $2.45$ $100$ $80$ $5.1$ $1.5828$ $1.5707$ $0.0047$ $3.94$ $120$ $120$ $4.7$ $3.8471$ $3.8331$ $0.0029$ $6.12$ $150$ $100$ $4.9$ $6.6608$ $6.6340$ $6.6340$ $7.34$ $150$ $120$ $6.2$ $8.7834$ $8.7580$ $0.0026$ $9.21$
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