Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization

Mathematical Programming

Volumn 134, Issue 1, 2012, Pages 223-257

  Bandeira, A S ^a   Scheinberg, K ^b   Vicente, L N ^c  

^a PRINCETON UNIVERSITY   (United States)

^b Lehigh University   (United States)

^c UNIVERSITY OF COIMBRA   (Portugal)

Author keywords

Compressed sensing; Derivative free optimization; Interpolation based trust region methods; L1 Minimization; Random sampling; Sparse recovery

Indexed keywords

COMPRESSIVE SENSING; DERIVATIVE-FREE OPTIMIZATION; RANDOM SAMPLING; SPARSE RECOVERY; TRUST-REGION METHODS;

IMAGE REGISTRATION; POLYNOMIALS; RECOVERY; SIGNAL RECONSTRUCTION;

INTERPOLATION;

EID: 84865690792     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-012-0578-z     Document Type: Conference Paper

References (31)

1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3-51 (2003)
2. Bandeira, A., Scheinberg, K., Vicente, L.N.: On partially sparse recovery. Tech. Rep. 11-13, Dept. Mathematics, Univ. Coimbra (2011)
3. Bandeira, A.S., Fickus, M., Mixon, D.G., Wong, P.: The road to deterministic matrices with the restricted isometry property. Submitted, available online: ArXiv:1202.1234 [math.FA] (2012)
4. Baraniuk, R.,Davenport, M.,DeVore, R.,Wakin,M.:Asimple proof of the restricted isometry property for random matrices. Constr. Approx. 28, 253-263 (2008)
5. Candès, E., Tao, T.: Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inf. Theory 52, 5406-5425 (2006)
6. Candès, E.J.: The restricted isometry property and its implications for compressed sensing. Compte Rendus de l'Academie des Sciences, Paris, Serie I, pp. 589-592 (2009)
7. Colson, B., Toint, Ph. L.: Optimizing partially separable functions without derivatives Optim.Methods Softw. 20, 493-508 (2005)
8. Conn, A.R., Gould, N.I.M., Toint, Ph.L.: Trust-Region Methods. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2000)
9. Conn, A.R., Scheinberg, K., Toint, Ph. L.: A derivative free optimization algorithm in practice. In: Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, Missouri, Sept 2-4 (1998)
10. Conn, A.R., Scheinberg, K., Vicente, L.N.: Global convergence of general derivative-free trust-region algorithms to first and second order critical points SIAM J. Optim. 20, 387-415 (2009)
11. Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2009)
12. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles Math. Program. 91, 201-213 (2002)
13. Dolan, E.D., Moré, J.J., Munson, T.S.: Optimality measures for performance profiles SIAM J. Optim. 16, 891-909 (2006)
14. Fasano, G., Morales, J.L., Nocedal, J.: On the geometry phase in model-based algorithms for derivative-free optimization Optim. Methods Softw. 24, 145-154 (2009)
15. Ferris, M.C., Deng, G.: Classification-based global search: An application to a simulation for breast cancer. In: Proceedings of 2008 NSF Engineering Research and Innovation Conference. Knoxville, Tennessee (2008)
16. Gould, N.I.M., Orban, D., Toint, Ph.L.: CUTEr (and SifDec), a constrained and unconstrained testing environment, revisited ACM Trans. Math. Softw. 29, 373-394 (2003)
17. Gratton, S., Toint, Ph.L., Tröltzsch, A.: An active-set trust-region method for derivative-free nonlinear bound-constrained optimization Optim. Methods Softw. 26, 873-894 (2011)
18. Jacques, L.:Ashort note on compressed sensing with partially known signal support Signal Process. 90, 3308-3312 (2010)
19. Moré, J.J., Sorensen, D.C.: Computing a trust region step SIAM J. Sci. Comput. 4, 553-572 (1983)
20. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)
21. Powell, M.J.D.: On trust region methods for unconstrained minimization without derivatives Math. Program. 97, 605-623 (2003)
22. Powell, M.J.D.: Least Frobenius norm updating of quadratic models that satisfy interpolation conditions Math. Program. 100, 183-215 (2004)
23. Powell, M.J.D.: The NEWUOA software for unconstrained optimization without derivatives. In: Nonconvex Optim. Appl., vol. 83, pp. 255-297. Springer, Berlin (2006)
24. Rauhut, H.: Compressive sensing and structured random matrices. In: Fornasier, M. (ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Series Comp. Appl.Math., pp. 1-92 (2010)
25. Rauhut, H., Ward, R.: Sparse Legendre expansions via -1-minimization. J. Approx. Theory 164, 517-533 (2012)
26. Scheinberg, K., Toint, P.L.: Self-correcting geometry in model-based algorithms for derivative-free unconstrained optimization SIAM J. Optim. 20, 3512-3532 (2010)
27. Tao, T.: Open question: Deterministic UUP matrices: Http://terrytao. wordpress.com/2007/07/02/open-question-deterministic-uup-matrices (2007)
28. Vaswani, N., Lu,W.: Modified-CS: Modifying compressive sensing for problems with partially known support IEEE Trans. Signal Process. 58, 4595-4607 (2010)
29. Wild, S.M.: MNH: A derivative-free optimization algorithm using minimal norm Hessians. In: Tenth Copper Mountain Conference on Iterative Methods (2008)
30. Wojtaszczyk, P.: Stability and instance optimality for gaussian measurements in compressed sensing Found. Comput. Math. 10, 1-13 (2010)
31. Zhang, Y.: Solving large-scale linear programs by interior-oint methods under the MATLAB environment Optim. Methods Softw. 10, 1-31 (1998)