Advertisement

Invex optimization revisited

  • Ksenia Bestuzheva
  • Hassan Hijazi
Article
  • 126 Downloads

Abstract

Given a non-convex optimization problem, we study conditions under which every Karush–Kuhn–Tucker (KKT) point is a global optimizer. This property is known as KT-invexity and allows to identify the subset of problems where an interior point method always converges to a global optimizer. In this work, we provide necessary conditions for KT-invexity in n dimensions and show that these conditions become sufficient in the two-dimensional case. As an application of our results, we study the Optimal Power Flow problem, showing that under mild assumptions on the variables’ bounds, our new necessary and sufficient conditions are met for problems with two degrees of freedom.

Keywords

Convex optimization Invex optimization Boundary-invexity Optimal power flow 

Notations

\(\partial S\)

Boundary of a set S.

\(x_i\)

ith component of vector \(\mathbf {x}\).

\(f_{x_{i}}^{\prime } = \frac{\partial f}{\partial x_i}\)

Partial derivative of f with respect to \(x_i\).

\(||\mathbf {x}||\)

Euclidean norm of vector \(\mathbf {x}\).

\(\mathbf {x} \cdot \mathbf {y}\)

The dot product of vectors \(\mathbf {x}\) and \(\mathbf {y}\).

\(\mathbf {x}^T\)

The transpose of vector \(\mathbf {x}\).

\(\overline{AB}\)

A segment between points A and B.

\(2\mathbb {N}, ~2\mathbb {N}{+}1\)

The sets of even and odd numbers.

\(f'_-(x), f'_+(x)\)

Left and right derivatives of f.

sign(x)

The sign function.

References

  1. 1.
    Abbena, E., Salamon, S., Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, Boca Raton (2006)zbMATHGoogle Scholar
  2. 2.
    Antczak, T.: (p, r)-invex sets and functions. J. Math. Anal. Appl. 263(2), 355–379 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bector, C., Singh, C.: B-vex functions. J. Optim. Theory Appl. 71(2), 237–253 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ben-Israel, A., Mond, B.: What is invexity? ANZIAM J. 28(1), 1–9 (1986)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Coffrin, C., Hijazi, H.L., Hentenryck, P.V.: The QC relaxation: a theoretical and computational study on optimal power flow. IEEE Trans. Power Syst. 31(4), 3008–3018 (2016).  https://doi.org/10.1109/TPWRS.2015.2463111 CrossRefGoogle Scholar
  7. 7.
    Craven, B.: Invex functions and constrained local minima. Bull. Austral. Math. Soc. 24(03), 357–366 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Craven, B.: Global invexity and duality in mathematical programming. Asia-Pac. J. Oper. Res. 19(2), 169 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Craven, B., Glover, B.: Invex functions and duality. J. Austral. Math. Soc. (Ser. A) 39(01), 1–20 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hanson, M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80(2), 545–550 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jeyakumar, V., Mond, B.: On generalised convex mathematical programming. J. Austral. Math. Soc. Ser. B Appl. Math. 34(01), 43–53 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Lehmann, K., Grastien, A., Hentenryck, P.V.: AC-feasibility on tree networks is NP-hard. IEEE Trans. Power Syst. 31(1), 798–801 (2016).  https://doi.org/10.1109/TPWRS.2015.2407363 CrossRefGoogle Scholar
  14. 14.
    Mangasarian, O.L.: Pseudo-convex functions. J. Soc. Ind. Appl. Math. Ser. A Control 3(2), 281–290 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Martin, D.: The essence of invexity. J. Optim. Theory Appl. 47(1), 65–76 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mendelson, B.: Introduction to Topology. Courier Corporation, New York (1990)zbMATHGoogle Scholar
  17. 17.
    Momoh, J., Adapa, R., El-Hawary, M.: A review of selected optimal power flow literature to 1993. i. Nonlinear and quadratic programming approaches. IEEE Trans. Power Syst. 14(1), 96–104 (1999).  https://doi.org/10.1109/59.744492 CrossRefGoogle Scholar
  18. 18.
    Momoh, J., El-Hawary, M., Adapa, R.: A review of selected optimal power flow literature to 1993. ii. Newton, linear programming and interior point methods. IEEE Trans. Power Syst. 14(1), 105–111 (1999).  https://doi.org/10.1109/59.744495 CrossRefGoogle Scholar
  19. 19.
    Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  20. 20.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  21. 21.
    Norden, A.P.: Theory of Surfaces, vol. 8. Gostekhizdat, Moscow (1956). (in Russian)Google Scholar
  22. 22.
    Pardalos, P.M., Schnitger, G.: Checking local optimality in constrained quadratic programming is NP-hard. Oper. Res. Lett. 7(1), 33–35 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Simmons, G.F.: Introduction to Topology and Modern Analysis. McGraw-Hill, Tokyo (1963). (The newest version was published by Krieger Publishing Company in 2003)Google Scholar
  24. 24.
    Verma, A.: Power grid security analysis: An optimization approach. Ph.D. thesis, Columbia University (2009)Google Scholar
  25. 25.
    Wang, Z., Fang, S.C., Xing, W.: On constraint qualifications: motivation, design and inter-relations. J. Ind. Manag. Optim. 9(4), 983–1001 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wrede, R.C., Spiegel, M.R.: Advanced Calculus. McGraw-Hill, New York (2010)Google Scholar
  27. 27.
    Xia, Y., Wang, S., Sheu, R.L.: S-lemma with equality and its applications. Math. Program. 156(1), 513–547 (2016).  https://doi.org/10.1007/s10107-015-0907-0 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Australian National University, Data61, CSIROActonAustralia
  2. 2.Los Alamos National LaboratoryLos AlamosUSA

Personalised recommendations