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Solving DC programs using the cutting angle method

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Abstract

In this paper, we propose a new algorithm for global minimization of functions represented as a difference of two convex functions. The proposed method is a derivative free method and it is designed by adapting the extended cutting angle method. We present preliminary results of numerical experiments using test problems with difference of convex objective functions and box-constraints. We also compare the proposed algorithm with a classical one that uses prismatical subdivisions.

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References

  1. Alexandrov, A.D.: On surfaces which may be represented by a difference of convex functions (in russian). Izvestia Akademii Nauk Kazakhskoj SSR, Seria Fiziko-Matematicheskikh Nauk 3, 3–20 (1949)

    Google Scholar 

  2. Bagirov, A.M., Rubinov, A.M.: Global minimization of increasing positively homogeneous functions over the unit simplex. Ann. Oper. Res. 98(1–4), 171–187 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bagirov, A.M., Rubinov, A.M.: Modified versions of the cutting angle method. In: Hadjisavvas, N., Pardalos, P.M. (eds.) Convex Analysis and Global Optimization, Nonconvex Optimization and Its Applications, vol. 54, pp. 245–268. Kluwer, Dordrecht (2001)

    Chapter  Google Scholar 

  4. Batten, L.M., Beliakov, G.: Fast algorithm for the cutting angle method of global optimization. J. Global Optim. 24, 149–161 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Beliakov, G.: Geometry and combinatorics of the cutting angle method. Optimization 52(4–5), 379–394 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Beliakov, G.: The cutting angle method: a tool for constrained global optimization. Optim. Methods Softw. 19, 137–151 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Beliakov, G.: A review of applications of the cutting angle methods. In: Rubinov, A., Jeyakumar, V. (eds.) Continuous Optimization, pp. 209–248. Springer, New York (2005)

    Chapter  Google Scholar 

  8. Beliakov, G.: Extended cutting angle method of global optimization. Pac. J. Optim. 4(1), 153–175 (2008)

    MATH  MathSciNet  Google Scholar 

  9. Beliakov, G., Ting, K.M., Murshed, M., Rubinov, A.M., Bertoli, M.: Efficient serial and parallel implementations of the cutting angle method. In: Di Pillo, G. (ed.) High Performance Algorithms and Software for Nonlinear Optimization, pp. 57–74. Kluwer Academic Publishers, Dordrecht (2003)

    Chapter  Google Scholar 

  10. Bougeard, M.: Contribution à la théorie de Morse en dimension finie. PhD thesis, Université de Paris IX, Paris (1978)

  11. Cheney, E.W., Goldstein, A.A.: Newton’s method for convex programming and Tchebycheff approximation. Numer. Math. 1, 253–268 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ferrer, A.: Representation of a polynomial function as a difference of convex polynomials, with an application. Lect. Notes Econ. Math. Syst. 502, 189–207 (2001)

    Article  MathSciNet  Google Scholar 

  14. A, Ferrer: Applying global optimization to a problem in short-term hydrotermal scheduling. Nonconvex Optim. Appl. 77, 263–285 (2005)

    Article  MATH  Google Scholar 

  15. Hartman, P.: On functions representable as a difference of convex functions. Pac. J. Math. 9, 707–713 (1959)

    Article  MATH  Google Scholar 

  16. Hoai An, L.T., Dinh Tao, P.: The DC (difference of convex functions) Programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Holmberg, K., Tuy, H.: A production-transportation problem with stochastic demand and concave production costs. Math. Program. 85, 157–179 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 1st edn. Springer, Heilderberg (1990)

    Book  MATH  Google Scholar 

  19. Horst, R., Pardalos, P.M., Thoai, NgV: Introduction to Global Optimization, first edition edn. Kluwer Academic Publishers, Dordrecht (1995)

    MATH  Google Scholar 

  20. Kelley Jr, J.E.: The cutting-plane method for solving convex programs. J. Soc. Indust. Appl. Math. 8, 703–712 (1960)

    Article  MathSciNet  Google Scholar 

  21. Konno, H., Thach, P.T., Tuy, H.: Optimization on Low Rank Nonconvex Structures. Kluwer Academic Publishers, New York (1997)

    Book  MATH  Google Scholar 

  22. Landis, E.M.: On functions representable as the difference of two convex functions. Dokl. Akad. Nauk SSSR 80, 9–11 (1951)

    MATH  MathSciNet  Google Scholar 

  23. Penot, J.P., Bougeard, M.L.: Approximation and decomposition properties of some classes of locally d.c. functions. Math. Program. 41, 195–227 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pey-Chun, C., Hansen, P., Jaumard, B., Tuy, H.: Solution of the multisource weber and conditional weber problems by d.c. programming. Oper. Res. 46(4), 548–562 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rubinov, A.M.: Abstract Convexity and Global Optimization, Volume 44 of Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

  26. Strekalovsky, A., Tsevendorj, I.: Testing the \(\cal R\)-strategy for a reverse convex problem. J. Global Optim. 13, 61–74 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tuy, H.: Convex Analysis and Global Optimization, 1st edn. Kluwer Academic Publishers, Dordrescht (1998)

    Book  MATH  Google Scholar 

  28. Vial, J.-P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This research by Dr. Albert Ferrer was partially supported by the Ministry of Science and Technology (Project No. MTM2011-29064-C03-01), and by Dr. Adil Bagirov was supported under Australian Research Council’s Discovery Projects funding scheme (Project No. DP140103213). The authors would like to thank an anonymous referee and an Associate Editor for their comments that helped to improve the quality of the paper.

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Correspondence to Albert Ferrer.

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The research of this author has been partially supported by the Ministerio de Ciencia y Tecnología, Project MTM2011-29064-C03-01.

Appendix

Appendix

In Appendix we describe test problems used in the numerical experiments.

1.1 Test problems with non-Lipschitz objective functions

Problem 10.1

$$\begin{aligned} \begin{array}{r@{\quad }l} \hbox {minimize}&{} f(x,y):=-\sin (\sqrt{3x+2y+|x-y|}),\\ \hbox {subject to:} &{}0\le x\le 5,\\ &{}0\le y\le 5.\\ \end{array} \end{aligned}$$
(24)

The function \(5(x^2+y^2)\), allows us to obtain a d.c. representations of the objective function \(f\) so DCECAM can be applied:

$$\begin{aligned} f(x,y)=5(x^2+y^2)-(-f(x,y)+5(x^2+y^2)). \end{aligned}$$

The best solution found is \((0.29658\ldots , 0.62279\ldots )\) with \(f^*=-0.99999825\ldots \).

Problem 10.2

$$\begin{aligned} \begin{array}{c@{\quad }c} \hbox {minimize}&{} \varphi (x):=a\sqrt{|1-x|}+|2-x|^3,\\ \hbox {subject to:} &{}1\le x\le 3,\\ \end{array} \end{aligned}$$
(25)

with \(a=0,9\) and \(a=1.5\). Then we have the convex functions

$$\begin{aligned} \begin{array}{r@{\quad }c@{\quad }l} \varphi _1(x)&{} =&{} |2-x|^3, \\ \varphi _2(x) &{}= &{} -a\sqrt{|1-x|}. \end{array} \end{aligned}$$

that satisfy

$$\begin{aligned} \varphi (x)=\varphi _1(x) - \varphi _2(x). \end{aligned}$$

Problem 10.3

$$\begin{aligned} \begin{array}{c@{\quad }c} \hbox {minimize}&{} \phi (x):=-\log (x)+\min \left\{ \sqrt{|1-x|},(2-x)^3,\sqrt{|3-x|}\right\} \\ \hbox {subject to:} &{}1\le x\le 3,\\ \end{array} \end{aligned}$$
(26)

Then we have the convex functions

$$\begin{aligned} \begin{array}{r@{\quad }c@{\quad }l} \phi _1(x)&= 6x^2-12x+8+\max \{0,-x^3\}-\log (x), \end{array} \end{aligned}$$

and

$$\begin{aligned} \phi _2(x)= \max \left\{ \begin{array}{l} -\sqrt{|3-x|}+6x^2-12x+8+\max \{0,-x^3\},\\ -\sqrt{|1-x|}+6x^2-12x+8+\max \{0,-x^3\},\\ \max \{0,x^3\}) \end{array}\right\} \end{aligned}$$

that satisfy

$$\begin{aligned} \phi (x)=\phi _1(x) - \phi _2(x). \end{aligned}$$

Problem 10.4

By combining the above-mentioned functions we can obtain new function with \(n\) variables.

$$\begin{aligned} \begin{array}{c@{\quad }c} \hbox {minimize}&{} f(x):=\sum ^n_{i=1}\varphi (x_i)\\ \hbox {subject to:} &{}1\le x_i\le 3, i=1,\ldots ,n\\ \end{array} \end{aligned}$$
(27)

and

$$\begin{aligned} \begin{array}{c@{\quad }c} \hbox {minimize}&{} f(x):=\sum ^n_{i=1}\phi (x_i)\\ \hbox {subject to:} &{}1\le x_i\le 3, i=1,\ldots ,n.\\ \end{array} \end{aligned}$$
(28)

1.2 Test problems with Lipschitz objective functions

By using the convex function \(g(x):=K\Vert x\Vert ^2\) with \(K>0\), we can obtain a DC representation of the objective functions of the test problems as follows.

$$\begin{aligned} f(x)=\left( f(x)+g(x)\right) -g(x), \end{aligned}$$
(29)

with \(K\) being a real number such that \(f(x)+k\sum _{j=1}^n x_j^2\) is a convex function.

Problem 10.5

The class of test problems \(HPTnXmY\).

The following class of test problems can be found in [19]:

$$\begin{aligned} \begin{array}{r@{\quad }l} \hbox {minimize}&{}-\sum _{i=1}^m 1/\left( \Vert x-a^i\Vert ^2+c_i\right) \\ \hbox {subject to}&{}x \in \mathrm{I\!R}^n,0\le x_j \le 10, j=1,\dots ,n\\ \end{array} \end{aligned}$$
(30)

where \(a^i \in \{x \in \mathrm{I\!R}^n:0\le x_j \le 10, 1\le j \le n\}\) and \(c_i > 0\). By using the convex function \(k\left( \sum _{j=1}^nx_j^2\right) \) with \(k>0\), we can obtain a d.c. representation of the objective function in (30) as follows. Consider \( f(x)=\sum _{i=1}^mf_i(x), \) with \(f_i(x):=1/\left( \Vert x-a^i\Vert ^2+c_i\right) \) and \(x \in \mathrm{I\!R}^n\). Hence, we can write

$$\begin{aligned} f(x)=\left( f(x)+k\sum \limits _{j=1}^nx_j^2\right) -\left( k\sum \limits _{j=1}^nx_j^2\right) , \end{aligned}$$
(31)

with \(k\) a real number such that \(f(x)+k\sum _{j=1}^nx_j^2\) is a convex function. The different instances of the test problem (30) are denoted by \(HPTnXmY\) where \(X\) represents the dimension and \(Y\) means the number of local optimal solutions of the instance. Parameters are given in Table 8.

Table 8 Parameters for the test problem \(HPTnXmY\)

Problem 10.6

The class of test problems \(TnXrY\).

Let \(x\in \mathrm{I\!R}^n\) be \(x=(x_1,\ldots ,x_n)\). A reduced version of the test problem

$$\begin{aligned} \begin{array}{r@{\quad }l} \hbox {minimize}&{}f(x)=\Pi _{i=1}^n(x_i^2+c_ix_i)\\ \hbox {subject to}&{}-2 \le x_i \le 1, i=1,\ldots ,n,\\ \end{array} \end{aligned}$$
(32)

where \(A\in \mathrm{I\!R}^{m*n}\) and \(b\in \mathrm{I\!R}^m\), can be found in [27]. The names of the different instances of the test problem (32) are denoted by \(TnXrY\), where \(X\) is the dimension and \(Y\) means the number of linear constraints of the instance. For numerical tests, we have chosen the instance \(Tn2r4\) with the parameters \(c_1=0.09\) and \(c_2=0.1\). As before, by using the convex function \(k\left( \sum _{j=1}^nx_j^2\right) \) different d.c. representations of the objective function can be obtained in the form:

$$\begin{aligned} f(x)=\left( f(x)+k\sum \limits _{j=1}^nx_j^2\right) -\left( k\sum \limits _{j=1}^nx_j^2\right) . \end{aligned}$$

We consider the values \(k=7.5\), \(k=8\) and \(k=8.5\).

Problem 10.7

The class of test problems \(HPBr1\).

The problem

$$\begin{aligned} \begin{array}{r@{\quad }l} \hbox {minimize}&{}f(x,y) = \frac{1}{4}(x+y)^2-\frac{1}{4}(x-y)^2\\ \hbox {subject to:} &{}-2\le x\le 3,\\ &{}-3\le y\le 4,\\ \end{array} \end{aligned}$$
(33)

which will be denoted by \(HPBr1\), is a nonconvex programming problem. The objective function of \(HPBr1\) is an homogeneous polynomial of degree two with two variables (in this case it is a hyperbole). It is known that the d.c. representation of \(xy\) in (33) is the optimal. Alternative non-optimal d.c. representations of \(xy\) are:

  1. (1)

    \(xy=\frac{1}{2}(x+y)^2-\frac{1}{2}(x^2+y^2)\), and

  2. (2)

    \(xy=\frac{1}{2}(x^2+y^2)-\frac{1}{2}(x-y)^2\).

Problem 10.8

The class of test problems \(COSr0\)

The problem

$$\begin{aligned} \begin{array}{r@{\quad }l} \hbox {minimize}&{}f(x,y):=0.03(x^2+y^2)-cos(x)cos(y)\\ \hbox {subject to:} &{}-6\le x\le 4,\\ &{}-5\le y\le 2,\\ \end{array} \end{aligned}$$
(34)

which will be denoted by \(COSr0\), is a multiextremal programming problem with minimizer \((0,0)\) and minimum \(-1\). The function \(k(x^2+y^2)\), \(k>0\) allows us to obtain many different d.c. representations of the objective function \(f\):

$$\begin{aligned} f(x,y)=(f(x,y)+k(x^2+y^2))-k(x^2+y^2). \end{aligned}$$

Problem 10.9

$$\begin{aligned} \begin{array}{r@{\quad }l} \hbox {minimize}&{}f(x):= f_1(x) - f_2(x),~x \in \mathrm{I\!R}^4\\ \hbox {subject to:} &{}-10 \le x_i \le 10,~i=1,2,3,4.\\ \end{array} \end{aligned}$$
(35)

Here

$$\begin{aligned} f_1(x) = | x_1-1|+200 \max \{0, | x_1| - x_2\} + 180 \max \{0, | x_3| - x_4\} + |x_3-1| \end{aligned}$$
$$\begin{aligned} +10.1 (|x_2-1| + |x_4-1|) + 4.95 |x_2+x_4-2|, \end{aligned}$$
$$\begin{aligned} f_2(x) = 100( | x_1| - x_2) + 90 ( | x_3| - x_4) + 4.95|x_2-x_4|. \end{aligned}$$

Problem 10.10

$$\begin{aligned} \begin{array}{r@{\quad }l} \hbox {minimize}&{}f(x):= f_1(x) - f_2(x),~x \in \mathrm{I\!R}^n\\ \hbox {subject to:} &{}-10 \le x_i \le 10,~i=1,\ldots ,n.\\ \end{array} \end{aligned}$$
(36)

Here

$$\begin{aligned} f_1(x) = | x_1 - 1| + 200 \sum \limits _{i=2}^n \max \{0, | x_{i-1}| - x_i\}, \end{aligned}$$
$$\begin{aligned} f_2(x) = 100 \sum \limits _{i=2}^n ( | x_{i-1}| - x_i). \end{aligned}$$

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Ferrer, A., Bagirov, A. & Beliakov, G. Solving DC programs using the cutting angle method. J Glob Optim 61, 71–89 (2015). https://doi.org/10.1007/s10898-014-0159-1

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