American Institute of Mathematical Sciences

Advanced
September  2018, 8(3): 315-326. doi: 10.3934/naco.2018020

On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints

M. S. Lee 1,, , B. S. Goh 2, , H. G. Harno 3, and  K. H. Lim 4,

1. 

School of Economics and Management, Xiamen University Malaysia, Selangor, Malaysia
2. 

School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Australia
3. 

Department of Aerospace and Software Engineering, Gyeongsang National University, South Korea
4. 

Department of Electrical and Computer Engineering, Curtin University Malaysia, Sarawak, Malaysia

* Corresponding author

Received  April 2017 Revised  June 2017 Published  June 2018

Lagrange multipliers are usually used in numerical methods to solve equality constrained optimization problems. However, when the intersection between a search region for a current point and the feasible set defined by the equality constraints is empty, Lagrange multipliers cannot be used without additional conditions. To cope with this condition, a new method based on a two-phase approximate greatest descent approach is presented in this paper. In Phase-Ⅰ, an accessory function is used to drive a point towards the feasible set and the optimal point of an objective function. It has been observed that for some current points, it may be necessary to maximize the objective function while minimizing the constraint violation function in a current search region in order to construct the best numerical iterations. When the current point is close to or inside the feasible set and when optimality conditions are nearly satisfied, the numerical iterations are switched to Phase-Ⅱ. The Lagrange multipliers are defined and used in this phase. The approximate greatest descent method is then applied to minimize a merit function which is constructed from the optimality conditions. Results of numerical experiments are presented to show the effectiveness of the aforementioned two-phase method.

Keywords: Two-phase AGD method, nonlinear optimization, equality constraints, Lyapunov function, Lagrange multiplier.
Mathematics Subject Classification: Primary: 49M15, 65K10; Secondary: 90C30, 90C90.
Citation: M. S. Lee, B. S. Goh, H. G. Harno, K. H. Lim. On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 315-326. doi: 10.3934/naco.2018020
References:
[1]

I. Bongartz, A. R. Conn, N. I. M. Gould and P. L. Toint, CUTE: Constrained and unconstrained testing environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160. doi: 10.1145/200979.201043. Google Scholar

[2]

R. H. Bryd, F. Curtis and J. Nocedal, An inexact SQP method for equality constrained optimization, SIAM Journal of Optimization, 19 (2008), 351-369. doi: 10.1137/060674004. Google Scholar

[3]

R. H. Bryd, F. Curtis and J. Nocedal, An inexact Newton method for nonconvex equality constrained optimization, Mathematical Programming, 122 (2010), 273-299. doi: 10.1007/s10107-008-0248-3. Google Scholar

[4]

A. R. Conn, N. I. M. Gould and P. L. Toint, Trust Region Methods, SIAM, United States of America, 2000. doi: 10.1137/1.9780898719857. Google Scholar

[5]

A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968. Google Scholar

[6]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Mathematical Programming, 91 (2002), 239-269. doi: 10.1007/s101070100244. Google Scholar

[7]

P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, Computing forward-difference intervals for numerical optimization, Society for Industrial and Applied Mathematics, 4 (1983), 310-321. doi: 10.1137/0904025. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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 (ed. H. Xu and X. Wang), Springer Science+Business Media Dordrecht, 2014. doi: 10.1007/978-94-017-8044-5_1. Google Scholar

[12]

N. I. M. Gould, D. Orban and P. L. Toint, CUTEr, a constrained and unconstrained testing environment (revisited), ACM Transactions on Mathematical Software, 29 (2003), 373-394. Google Scholar

[13]

H. Z. Hassan, A. A. Mohamad and G. E. Atteia, An algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy, Journal of Computational and Applied Mathematics, 236 (1997), 2622-2631. doi: 10.1016/j.cam.2011.12.019. Google Scholar

[14]

Y. C. Ho, Differential games, dynamic optimization and generalized control theory, Journal of Optimization Theory and Applications, 6 (1970), 179-209. doi: 10.1007/BF00926600. Google Scholar

[15]

J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. Google Scholar

[16]

C. Li, Z. Wang and D. Zhu, A line search filter inexact reduced Hessian method for nonlinear equality constrained optimization, Journal of Applied Mathematics and Computing, 48 (2015), 365-380. doi: 10.1007/s12190-014-0807-0. Google Scholar

[17]

R. S. Stepleman and N. D. Winarsky, Adaptive numerical differenctiation, Mathematics of Computation, 33 (1979), 1257-1264. doi: 10.1090/S0025-5718-1979-0537969-8. Google Scholar

[18]

G. W. Stewart, A modification of Davidon's minimization method to accept difference approximation of derivatives, Journal of Association for Computing Machinery, 14 (1967), 72-83. doi: 10.1145/321371.321377. Google Scholar

[19]

A. Vardi, A trust region algorithm for equality constrained minimization, SIAM Journal of Optimization, 22 (1983), 575-591. doi: 10.1137/0722035. Google Scholar

[20]

T. L. Vincent and G. Leitmann, Control space properties of cooperative games, Journal of Optimization Theory and Applications, 6 (1970), 91-116. doi: 10.1007/BF00927045. Google Scholar

[21]

Y. Yuan, Recent advances in trust region algorithms, Mathematical Programming, 151 (2015), 249-281. doi: 10.1007/s10107-015-0893-2. Google Scholar

[22]

L. A. Zadeh, Optimality and non-scalar-valued performance criteria, IEEE Transactions on Automatic Control, 8 (1963), 59-60. doi: 10.1109/TAC.1963.1105511. Google Scholar

[23]

X. Zhu and D. Pu, A restoration-free filter SQP algorithm for equality constrained optimization, Applied Mathematics and Computation, 219 (2013), 6016-6029. doi: 10.1016/j.amc.2012.12.002. Google Scholar

show all references

References:
[1]

I. Bongartz, A. R. Conn, N. I. M. Gould and P. L. Toint, CUTE: Constrained and unconstrained testing environment, ACM Transactions on Mathematical Software, 21 (1995), 123-160. doi: 10.1145/200979.201043. Google Scholar

[2]

R. H. Bryd, F. Curtis and J. Nocedal, An inexact SQP method for equality constrained optimization, SIAM Journal of Optimization, 19 (2008), 351-369. doi: 10.1137/060674004. Google Scholar

[3]

R. H. Bryd, F. Curtis and J. Nocedal, An inexact Newton method for nonconvex equality constrained optimization, Mathematical Programming, 122 (2010), 273-299. doi: 10.1007/s10107-008-0248-3. Google Scholar

[4]

A. R. Conn, N. I. M. Gould and P. L. Toint, Trust Region Methods, SIAM, United States of America, 2000. doi: 10.1137/1.9780898719857. Google Scholar

[5]

A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968. Google Scholar

[6]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Mathematical Programming, 91 (2002), 239-269. doi: 10.1007/s101070100244. Google Scholar

[7]

P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, Computing forward-difference intervals for numerical optimization, Society for Industrial and Applied Mathematics, 4 (1983), 310-321. doi: 10.1137/0904025. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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 (ed. H. Xu and X. Wang), Springer Science+Business Media Dordrecht, 2014. doi: 10.1007/978-94-017-8044-5_1. Google Scholar

[12]

N. I. M. Gould, D. Orban and P. L. Toint, CUTEr, a constrained and unconstrained testing environment (revisited), ACM Transactions on Mathematical Software, 29 (2003), 373-394. Google Scholar

[13]

H. Z. Hassan, A. A. Mohamad and G. E. Atteia, An algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy, Journal of Computational and Applied Mathematics, 236 (1997), 2622-2631. doi: 10.1016/j.cam.2011.12.019. Google Scholar

[14]

Y. C. Ho, Differential games, dynamic optimization and generalized control theory, Journal of Optimization Theory and Applications, 6 (1970), 179-209. doi: 10.1007/BF00926600. Google Scholar

[15]

J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. Google Scholar

[16]

C. Li, Z. Wang and D. Zhu, A line search filter inexact reduced Hessian method for nonlinear equality constrained optimization, Journal of Applied Mathematics and Computing, 48 (2015), 365-380. doi: 10.1007/s12190-014-0807-0. Google Scholar

[17]

R. S. Stepleman and N. D. Winarsky, Adaptive numerical differenctiation, Mathematics of Computation, 33 (1979), 1257-1264. doi: 10.1090/S0025-5718-1979-0537969-8. Google Scholar

[18]

G. W. Stewart, A modification of Davidon's minimization method to accept difference approximation of derivatives, Journal of Association for Computing Machinery, 14 (1967), 72-83. doi: 10.1145/321371.321377. Google Scholar

[19]

A. Vardi, A trust region algorithm for equality constrained minimization, SIAM Journal of Optimization, 22 (1983), 575-591. doi: 10.1137/0722035. Google Scholar

[20]

T. L. Vincent and G. Leitmann, Control space properties of cooperative games, Journal of Optimization Theory and Applications, 6 (1970), 91-116. doi: 10.1007/BF00927045. Google Scholar

[21]

Y. Yuan, Recent advances in trust region algorithms, Mathematical Programming, 151 (2015), 249-281. doi: 10.1007/s10107-015-0893-2. Google Scholar

[22]

L. A. Zadeh, Optimality and non-scalar-valued performance criteria, IEEE Transactions on Automatic Control, 8 (1963), 59-60. doi: 10.1109/TAC.1963.1105511. Google Scholar

[23]

X. Zhu and D. Pu, A restoration-free filter SQP algorithm for equality constrained optimization, Applied Mathematics and Computation, 219 (2013), 6016-6029. doi: 10.1016/j.amc.2012.12.002. Google Scholar

Figure 1.  Level curves of an objective function $f(x_1,x_2) = x_1+x^{2}_2$ which is minimized and is subjected to an equality constraint $x_1^2+x_2^2 = 1$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Comparison of CPU time (second) between SQP and AGD
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Comparison of number of iteration between SQP and AGD
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Comparison of SQP method and a two-phase AGD method (Parameters: $n$- number of variables, $m$-number of equations, $k$-number of iterations, $f(x^*)$-optimal value)
Problem Dim. SQP Two-Phase AGD
$n$ $m$ $k$ $f(x^*)$ CPU Time (second) $k$ $f(x^*)$ CPU Time (second)
BT02 3 1 19 $3.26\times 10^{-2}$ 0.9631 14 $3.26\times 10^{-2}$ 0.3323
BT03 5 3 8 $4.09\times 10^{0} $ 0.9505 10 $4.09\times 10^{0} $ 0.5181
BT04 3 2 13 $-4.55\times 10^{1} $ 0.9631 7 $-4.55\times 10^{1} $ 0.3484
BT05 3 2 7 $9.61\times10^{2}$ 0.9157 8 $9.61\times10^{2}$ 0.4478
BT06 5 2 19 $2.77\times 10^{-1} $ 0.9415 9 $2.77\times 10^{-1} $ 0.3677
BT09 4 2 12 $-1.00\times 10^{0} $ 1.0625 12 $-1.00\times 10^{0} $ 0.4392
BT10 2 2 6 $-1.00\times 10^{0} $ 0.8608 10 $-1.00\times 10^{0}$ 0.2908
BT11 5 3 11 $8.23\times 10^{-1} $ 0.9820 10 $8.23\times 10^{-1} $ 0.5528
BT12 5 3 8 $6.19\times 10^{0} $ 0.9406 24 $6.19\times 10^{0} $ 0.6012
HS06 2 1 9 $5.32\times 10^{-17} $ 0.8338 5 $3.84\times 10^{-10} $ 0.2483
HS07 2 1 9 $-1.73\times 10^{0} $ 0.9611 9 $ -1.73\times 10^{0} $ 0.3172
HS08 2 2 6 $-1.00\times 10^{0} $ 0.8552 5 $-1.00\times 10^{0} $ 0.2725
HS26 3 1 49 $1.94\times 10^{-11} $ 1.0768 12 $ 1.71\times 10^{-6} $ 0.3273
HS27 3 1 22 $4.00\times 10^{-2}$ 2.5163 12 $4.00\times 10^{-2}$ 0.3359
HS28 3 1 9 $6.42\times 10^{-16}$ 0.9985 5 $2.81\times 10^{-8} $ 0.3426
HS39 4 2 12 $-1.00\times 10^{0} $ 0.8979 12 $-1.00\times 10^{0} $ 0.3527
HS40 4 3 6 $-2.50\times 10^{-1}$ 0.9206 3 $-2.50\times 10^{-1}$ 0.3657
HS46 5 2 29 $5.11\times 10^{-11}$ 1.1155 8 $2.93\times 10^{-5}$ 0.4488
HS48 5 2 9 $6.14\times 10^{-13}$ 0.8890 3 $3.12\times 10^{-12}$ 0.3000
HS49 5 2 26 $2.36\times 10^{-9}$ 1.0728 10 $2.03\times 10^{-4}$ 0.4787
HS50 5 3 14 $2.10\times 10^{-14}$ 0.8918 9 $4.68\times 10^{-15}$ 0.3810
HS51 5 3 8 $7.39\times 10^{-17}$ 0.8689 4 $1.94\times 10^{-11}$ 0.3319
HS52 5 3 8 $5.33\times 10^{0}$ 0.8330 10 $5.33\times 10^{0}$ 0.4217
HS53 5 3 7 $4.09\times 10^{0}$ 0.9753 10 $4.09\times 10^{0}$ 0.4867
HS61 3 2 11 $-1.44\times 10^{2}$ 0.8812 7 $-1.44\times 10^{2}$ 0.3707
HS77 5 2 16 $2.42\times 10^{-1}$ 0.9192 11 $2.42\times 10^{-1}$ 0.4633
HS79 5 3 12 $7.88\times 10^{-2}$ 0.8923 5 $7.88\times 10^{-2}$ 0.3731
HS1001Lnp 7 2 16 $6.81\times 10^{2}$ 0.9920 11 $6.81\times 10^{2}$ 0.7957
maratos 2 1 3 $-1.00\times 10^{0} $ 0.8387 2 $-1.00\times 10^{0} $ 0.2440
mswright 5 3 19 $1.29\times 10^{0}$ 1.1139 13 $1.29\times 10^{0}$ 0.5045
Table Options

Download as excel

Problem Dim. SQP Two-Phase AGD
$n$ $m$ $k$ $f(x^*)$ CPU Time (second) $k$ $f(x^*)$ CPU Time (second)
BT02 3 1 19 $3.26\times 10^{-2}$ 0.9631 14 $3.26\times 10^{-2}$ 0.3323
BT03 5 3 8 $4.09\times 10^{0} $ 0.9505 10 $4.09\times 10^{0} $ 0.5181
BT04 3 2 13 $-4.55\times 10^{1} $ 0.9631 7 $-4.55\times 10^{1} $ 0.3484
BT05 3 2 7 $9.61\times10^{2}$ 0.9157 8 $9.61\times10^{2}$ 0.4478
BT06 5 2 19 $2.77\times 10^{-1} $ 0.9415 9 $2.77\times 10^{-1} $ 0.3677
BT09 4 2 12 $-1.00\times 10^{0} $ 1.0625 12 $-1.00\times 10^{0} $ 0.4392
BT10 2 2 6 $-1.00\times 10^{0} $ 0.8608 10 $-1.00\times 10^{0}$ 0.2908
BT11 5 3 11 $8.23\times 10^{-1} $ 0.9820 10 $8.23\times 10^{-1} $ 0.5528
BT12 5 3 8 $6.19\times 10^{0} $ 0.9406 24 $6.19\times 10^{0} $ 0.6012
HS06 2 1 9 $5.32\times 10^{-17} $ 0.8338 5 $3.84\times 10^{-10} $ 0.2483
HS07 2 1 9 $-1.73\times 10^{0} $ 0.9611 9 $ -1.73\times 10^{0} $ 0.3172
HS08 2 2 6 $-1.00\times 10^{0} $ 0.8552 5 $-1.00\times 10^{0} $ 0.2725
HS26 3 1 49 $1.94\times 10^{-11} $ 1.0768 12 $ 1.71\times 10^{-6} $ 0.3273
HS27 3 1 22 $4.00\times 10^{-2}$ 2.5163 12 $4.00\times 10^{-2}$ 0.3359
HS28 3 1 9 $6.42\times 10^{-16}$ 0.9985 5 $2.81\times 10^{-8} $ 0.3426
HS39 4 2 12 $-1.00\times 10^{0} $ 0.8979 12 $-1.00\times 10^{0} $ 0.3527
HS40 4 3 6 $-2.50\times 10^{-1}$ 0.9206 3 $-2.50\times 10^{-1}$ 0.3657
HS46 5 2 29 $5.11\times 10^{-11}$ 1.1155 8 $2.93\times 10^{-5}$ 0.4488
HS48 5 2 9 $6.14\times 10^{-13}$ 0.8890 3 $3.12\times 10^{-12}$ 0.3000
HS49 5 2 26 $2.36\times 10^{-9}$ 1.0728 10 $2.03\times 10^{-4}$ 0.4787
HS50 5 3 14 $2.10\times 10^{-14}$ 0.8918 9 $4.68\times 10^{-15}$ 0.3810
HS51 5 3 8 $7.39\times 10^{-17}$ 0.8689 4 $1.94\times 10^{-11}$ 0.3319
HS52 5 3 8 $5.33\times 10^{0}$ 0.8330 10 $5.33\times 10^{0}$ 0.4217
HS53 5 3 7 $4.09\times 10^{0}$ 0.9753 10 $4.09\times 10^{0}$ 0.4867
HS61 3 2 11 $-1.44\times 10^{2}$ 0.8812 7 $-1.44\times 10^{2}$ 0.3707
HS77 5 2 16 $2.42\times 10^{-1}$ 0.9192 11 $2.42\times 10^{-1}$ 0.4633
HS79 5 3 12 $7.88\times 10^{-2}$ 0.8923 5 $7.88\times 10^{-2}$ 0.3731
HS1001Lnp 7 2 16 $6.81\times 10^{2}$ 0.9920 11 $6.81\times 10^{2}$ 0.7957
maratos 2 1 3 $-1.00\times 10^{0} $ 0.8387 2 $-1.00\times 10^{0} $ 0.2440
mswright 5 3 19 $1.29\times 10^{0}$ 1.1139 13 $1.29\times 10^{0}$ 0.5045
[1]

Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial & Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749
[2]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
[3]

Feng Ma, Mingfang Ni. A two-phase method for multidimensional number partitioning problem. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 203-206. doi: 10.3934/naco.2013.3.203
[4]

Zhongwen Chen, Songqiang Qiu, Yujie Jiao. A penalty-free method for equality constrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (2) : 391-409. doi: 10.3934/jimo.2013.9.391
[5]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123
[6]

Tien-Fu Liang, Hung-Wen Cheng. Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method. Journal of Industrial & Management Optimization, 2011, 7 (2) : 365-383. doi: 10.3934/jimo.2011.7.365
[7]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401
[8]

Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010
[9]

Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019075
[10]

Feimin Huang, Dehua Wang, Difan Yuan. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3535-3575. doi: 10.3934/dcds.2019146
[11]

Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137
[12]

Jan Prüss, Jürgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5379-5405. doi: 10.3934/dcds.2013.33.5379
[13]

Marianne Korten, Charles N. Moore. Regularity for solutions of the two-phase Stefan problem. Communications on Pure & Applied Analysis, 2008, 7 (3) : 591-600. doi: 10.3934/cpaa.2008.7.591
[14]

T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665
[15]

Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325
[16]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1
[17]

Jie Jiang, Yinghua Li, Chun Liu. Two-phase incompressible flows with variable density: An energetic variational approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3243-3284. doi: 10.3934/dcds.2017138
[18]

V. S. Manoranjan, Hong-Ming Yin, R. Showalter. On two-phase Stefan problem arising from a microwave heating process. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1155-1168. doi: 10.3934/dcds.2006.15.1155
[19]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-18. doi: 10.3934/dcds.2019239
[20]

Mohammad Hassan Farshbaf-Shaker, Takeshi Fukao, Noriaki Yamazaki. Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier. Conference Publications, 2015, 2015 (special) : 418-427. doi: 10.3934/proc.2015.0418

 Impact Factor: 

Tools

Metrics

  • PDF downloads (73)
  • HTML views (135)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences