# Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions

- 41 Downloads

## Abstract

We developed a long-step path-following algorithm for a class of symmetric programming problems with nonlinear convex objective functions. The theoretical framework is developed for functions compatible in the sense of Nesterov and Nemirovski with \(-\,\ln \det \) barrier function. Complexity estimates similar to the case of a linear-quadratic objective function are established, which gives an upper bound for the total number of Newton steps. The theoretical scheme is implemented for a class of spectral objective functions which includes the case of quantum (von Neumann) entropy objective function, important from the point of view of applications. We explicitly compare our numerical results with the only known competitor.

## Keywords

Convex optimization Symmetric programming Nonlinear objective functions Self-concordance Interior-point methods Matrix monotonicity Von Neumann entropy## Notes

### Acknowledgements

This research is supported in part by Simmons Foundation Grant 275013.

## References

- 1.ApS, MOSEK: The MOSEK optimization toolbox for MATLAB manual, version 8.0 (Revision 60). http://docs.mosek.com/8.0/toolbox/index.html (2017)
- 2.Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefGoogle Scholar
- 3.Chandrasecaran, V., Shah, P.: Relative entropy optimization and its applications. Math. Program. Ser. A
**161**(1–2), 1–32 (2017)MathSciNetCrossRefGoogle Scholar - 4.den Hertog, D.: Interior Point Approach to Linear. Quadratic and Convex Programming. Springer, Dordrecht (1994)CrossRefGoogle Scholar
- 5.den Hertog, D., Roos, C., Terlaky, T.: On the classical logarithmic barrier function method for a class of smooth convex programming problems. J. Optim. Theory Appl.
**73**(1), 1–25 (1992)MathSciNetCrossRefGoogle Scholar - 6.Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
- 7.Fawzi, H., Saunderson, J., Parrilo, P.A.: Semidefinite approximations of the matrix logarithm. Comput. Math. Found. (2018). https://doi.org/10.1007/s10208-018-9385-0
- 8.Faybusovich, L.: Semidefinite programming: a path-following algorithm for a linear-quadratic functional. SIAM J. Optim.
**6**(4), 1007–1024 (1996)MathSciNetCrossRefGoogle Scholar - 9.Faybusovich, L.: Several Jordan-algebraic aspects of optimization. Optimization
**57**(3), 379–393 (2008)MathSciNetCrossRefGoogle Scholar - 10.Faybusovich, L.: On Hazan’s algorithm for symmetric programming problems. J. Optim. Theory Appl.
**164**(3), 915–932 (2015)MathSciNetCrossRefGoogle Scholar - 11.Faybusovich, L.: Primal-dual potential reduction algorithm for symmetric programming problems with nonlinear objective functions. Linear Algebra Appl.
**536**, 228–249 (2018)MathSciNetCrossRefGoogle Scholar - 12.Faybusovich, L., Tsuchiya, T.: Matrix monotonicity and self-concordance: How to handle quantum entropy in optimization problems. Optim. Lett.
**11**(8), 1513–1526 (2017)MathSciNetCrossRefGoogle Scholar - 13.Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx (2017)
- 14.Korányi, A.: Monotone functions on formally real Jordan algebras. Math. Ann.
**269**, 73–76 (1984)MathSciNetCrossRefGoogle Scholar - 15.Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB, In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)Google Scholar
- 16.MathWorks, Inc.: MATLAB R2016a. Natick, MA (2016)Google Scholar
- 17.Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer, Boston (2004)CrossRefGoogle Scholar
- 18.Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
- 19.Sutter, D., Sutter, T., Esfahani, P.M., Renner, R.: Efficient approximation of quantum channel capacities. IEEE Trans. Inf. Theory
**62**(1), 578–598 (2016)MathSciNetCrossRefGoogle Scholar - 20.Toh, K.C., Todd, M.J., Tütüncü, R.H.: SDPT3—a matlab software package for semidefinite programming. Optim. Methods Softw.
**11**(1–4), 545–581 (1999)MathSciNetCrossRefGoogle Scholar - 21.Zinchenko, Y., Friedland, S., Gour, G.: Numerical estimation of the relative entropy of entanglement. Phys. Rev. A
**82**(5), 052336 (2010)CrossRefGoogle Scholar