A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited.
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Andreani, R., Martínez, J.M., Schuverdt, M.L.: On the relation between the Constant Positive Linear Dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125, 473–485 (2005)
Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On Augmented Lagrangian Methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007)
Andreani, R., Martínez, J.M., Schuverdt, M.L.: On second-order optimality conditions for nonlinear programming. Optimization 56, 529–542 (2007)
Andretta, M., Birgin, E.G., Martínez, J.M.: Practical active-set Euclidean trust-region method with spectral projected gradients for bound-constrained minimization. Optimization 54, 305–325 (2005)
Anitescu, M.: Degenerate nonlinear programming with a quadratic growth condition. SIAM J. Optim. 1116–1135 (2000)
Birgin, E.G., Martínez, J.M.: A box constrained optimization algorithm with negative curvature directions and spectral projected gradients. Computing (Suppl.) 15, 49–60 (2001)
Birgin, E.G., Martínez, J.M.: Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput. Optim. Appl. 23, 101–125 (2002)
Birgin, E.G., Martínez, J.M.: Structured minimal-memory inexact quasi-Newton method and secant preconditioners for Augmented Lagrangian optimization. Comput. Optim. Appl. 39, 1–16 (2008)
Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)
Birgin, E.G., Martínez, J.M., Raydan, M.: Algorithm 813: SPG—Software for convex-constrained optimization. ACM Trans. Math. Softw. 27, 340–349 (2001)
Birgin, E.G., Martínez, J.M., Raydan, M.: Inexact Spectral Projected Gradient methods on convex sets. IMA J. Numer. Anal. 23, 539–559 (2003)
Byrd, R.H., Schnabel, R.B., Shultz, G.A.: A trust region algorithm for nonlinearly constrained optimization. SIAM J. Numer. Anal. 24, 1152–1170 (1987)
Coleman, T.F., Liu, J., Yuan, W.: A new trust-region algorithm for equality constrained optimization. Comput. Optim. Appl. 21, 177–199 (2002)
Conn, A.R., Gould, N.I.M., Orban, D., Toint, Ph.L.: A primal-dual trust region algorithm for non-convex nonlinear programming. Math. Program. 87, 215–249 (2000)
Conn, A.R., Gould, N.I.M., Toint, Ph.L.: Trust Region Methods. MPS/SIAM Series on Optimization. SIAM, Philadelphia (2000)
Dennis, J.E., Vicente, L.N.: On the convergence theory of trust-region-based algorithms for equality-constrained optimization. SIAM J. Optim. 7, 527–550 (1997)
Dennis, J.E., Heinkenschloss, M., Vicente, L.N.: Trust-region interior-point SQP algorithms for a class of nonlinear programming problems. SIAM J. Control Optim. 36, 1750–1794 (1998)
Di Pillo, G., Lucidi, S., Palagi, L.: Convergence to second-order stationary points of a primal-dual algorithm model for nonlinear programming. Math. Oper. Res. 30, 897–915 (2005)
Facchinei, F., Lucidi, S.: Convergence to second order stationary points in inequality constrained optimization. Math. Oper. Res. 23, 746–766 (1998)
Fletcher, R.: Practical Methods of Optimization. Wiley, New York (1987)
Forsgren, A., Murray, W.: Newton methods for large-scale linear inequality constrained problems. SIAM J. Optim. 7, 132–176 (1997)
Gould, N.I.M., Lucidi, S., Roma, M., Toint, Ph.L.: Exploiting negative curvature directions in linesearch methods for unconstrained optimization. Optim. Methods Softw. 14, 75–98 (2000)
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)
Hager, W.W., Zhang, H.C.: A new active set algorithm for box constrained optimization. SIAM J. Optim. 17, 526–557 (2006)
Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)
Levitin, E.S., Milyutin, A.A., Osmolovskii, N.P.: Higher order conditions for a local minimum in problems with constraints. Russian Math. Surv. 33, 97–168 (1978)
Mangasarian, O.L., Fromovitz, S.: The Fritz-John necessary optimality conditions in presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)
Mccormick, G.P.: A modification of Armijo’s step-size rule for negative curvature. Math. Program. 13, 111–115 (1977)
Moré, J.J., Sorensen, D.C.: On the use of directions of negative curvature in a modified Newton method. Math. Program. 16, 1–20 (1979)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)
Qi, L., Wei, Z.: On the constant positive linear dependence condition and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)
Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control Optim. 12, 268–285 (1974)
Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993)
Shultz, G.A., Schnabel, R.B., Byrd, R.H.: A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties. SIAM J. Numer. Anal. 22, 47–67 (1985)
Sorensen, D.C.: Newton’s method with a model trust region modification. SIAM J. Numer. Anal. 19, 409–426 (1982)
Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)
Zhang, J., Xu, C.: A class of indefinite dogleg path methods for unconstrained minimization. SIAM J. Optim. 9, 646–667 (1999)
This work was supported by PRONEX-Optimization (PRONEX—CNPq/FAPERJ E-26/171.510/2006—APQ1), FAPESP (Grants 2006/53768-0 and 2005/57684-2) and CNPq.
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Andreani, R., Birgin, E.G., Martínez, J.M. et al. Second-order negative-curvature methods for box-constrained and general constrained optimization. Comput Optim Appl 45, 209–236 (2010). https://doi.org/10.1007/s10589-009-9240-y
- Nonlinear programming
- Augmented Lagrangians
- Global convergence
- Optimality conditions
- Second-order conditions
- Constraint qualifications