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Invex optimization revisited

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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.

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Abbreviations

\(\partial S\) :

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. Abbena, E., Salamon, S., Gray, A.: Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, Boca Raton (2006)

    MATH  Google Scholar 

  2. Antczak, T.: (p, r)-invex sets and functions. J. Math. Anal. Appl. 263(2), 355–379 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bector, C., Singh, C.: B-vex functions. J. Optim. Theory Appl. 71(2), 237–253 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ben-Israel, A., Mond, B.: What is invexity? ANZIAM J. 28(1), 1–9 (1986)

    MathSciNet  MATH  Google Scholar 

  5. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  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

    Article  Google Scholar 

  7. Craven, B.: Invex functions and constrained local minima. Bull. Austral. Math. Soc. 24(03), 357–366 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Craven, B.: Global invexity and duality in mathematical programming. Asia-Pac. J. Oper. Res. 19(2), 169 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Craven, B., Glover, B.: Invex functions and duality. J. Austral. Math. Soc. (Ser. A) 39(01), 1–20 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hanson, M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80(2), 545–550 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jeyakumar, V., Mond, B.: On generalised convex mathematical programming. J. Austral. Math. Soc. Ser. B Appl. Math. 34(01), 43–53 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Springer, New York (2002)

    Book  MATH  Google Scholar 

  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

    Article  Google Scholar 

  14. Mangasarian, O.L.: Pseudo-convex functions. J. Soc. Ind. Appl. Math. Ser. A Control 3(2), 281–290 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  15. Martin, D.: The essence of invexity. J. Optim. Theory Appl. 47(1), 65–76 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mendelson, B.: Introduction to Topology. Courier Corporation, New York (1990)

    MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  19. Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  20. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  21. Norden, A.P.: Theory of Surfaces, vol. 8. Gostekhizdat, Moscow (1956). (in Russian)

    Google Scholar 

  22. Pardalos, P.M., Schnitger, G.: Checking local optimality in constrained quadratic programming is NP-hard. Oper. Res. Lett. 7(1), 33–35 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  24. Verma, A.: Power grid security analysis: An optimization approach. Ph.D. thesis, Columbia University (2009)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wrede, R.C., Spiegel, M.R.: Advanced Calculus. McGraw-Hill, New York (2010)

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. The Australian National University, Data61, CSIRO, Acton, Canberra, 2601, Australia

    Ksenia Bestuzheva & Hassan Hijazi

  2. Los Alamos National Laboratory, Los Alamos, NM, 87544, USA

    Hassan Hijazi

Authors
  1. Ksenia Bestuzheva
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  2. Hassan Hijazi
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Correspondence to Ksenia Bestuzheva.

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Bestuzheva, K., Hijazi, H. Invex optimization revisited. J Glob Optim 74, 753–782 (2019). https://doi.org/10.1007/s10898-018-0650-1

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  • DOI: https://doi.org/10.1007/s10898-018-0650-1

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