Actual source code: armijo.h
1: // Context for an Armijo (nonmonotone) linesearch for unconstrained
2: // minimization.
3: //
4: // Given a function f, the current iterate x, and a descent direction d:
5: // Find the smallest i in 0, 1, 2, ..., such that:
6: //
7: // f(x + (beta**i)d) <= f(x) + (sigma*beta**i)<grad f(x),d>
8: //
9: // The nonmonotone modification of this linesearch replaces the f(x) term
10: // with a reference value, R, and seeks to find the smallest i such that:
11: //
12: // f(x + (beta**i)d) <= R + (sigma*beta**i)<grad f(x),d>
13: //
14: // This modification does effect neither the convergence nor rate of
15: // convergence of an algorithm when R is chosen appropriately. Essentially,
16: // R must decrease on average in some sense. The benefit of a nonmonotone
17: // linesearch is that local minimizers can be avoided (by allowing increase
18: // in function value), and typically, fewer iterations are performed in
19: // the main code.
20: //
21: // The reference value is chosen based upon some historical information
22: // consisting of function values for previous iterates. The amount of
23: // historical information used is determined by the memory size where the
24: // memory is used to store the previous function values. The memory is
25: // initialized to alpha*f(x^0) for some alpha >= 1, with alpha=1 signifying
26: // that we always force decrease from the initial point.
27: //
28: // The reference value can be the maximum value in the memory or can be
29: // chosen to provide some mean descent. Elements are removed from the
30: // memory with a replacement policy that either removes the oldest
31: // value in the memory (FIFO), or the largest value in the memory (MRU).
32: //
33: // Additionally, we can add a watchdog strategy to the search, which
34: // essentially accepts small directions and only checks the nonmonotonic
35: // descent criteria every m-steps. This strategy is NOT implemented in
36: // the code.
37: //
38: // Finally, care must be taken when steepest descent directions are used.
39: // For example, when the Newton direction is not not satisfy a sufficient
40: // descent criteria. The code will apply the same test regardless of
41: // the direction. This type of search may not be appropriate for all
42: // algorithms. For example, when a gradient direction is used, we may
43: // want to revert to the best point found and reset the memory so that
44: // we stay in an appropriate level set after using a gradient steps.
45: // This type of search is currently NOT supported by the code.
46: //
47: // References:
48: // Armijo, "Minimization of Functions Having Lipschitz Continuous
49: // First-Partial Derivatives," Pacific Journal of Mathematics, volume 16,
50: // pages 1-3, 1966.
51: // Ferris and Lucidi, "Nonmonotone Stabilization Methods for Nonlinear
52: // Equations," Journal of Optimization Theory and Applications, volume 81,
53: // pages 53-71, 1994.
54: // Grippo, Lampariello, and Lucidi, "A Nonmonotone Line Search Technique
55: // for Newton's Method," SIAM Journal on Numerical Analysis, volume 23,
56: // pages 707-716, 1986.
57: // Grippo, Lampariello, and Lucidi, "A Class of Nonmonotone Stabilization
58: // Methods in Unconstrained Optimization," Numerische Mathematik, volume 59,
59: // pages 779-805, 1991.
61: #ifndef __TAO_ARMIJO_H
64: #include "src/tao_impl.h"
65: #include "tao_solver.h"
67: typedef struct {
68: double *memory;
70: double alpha; // Initial reference factor >= 1
71: double beta; // Steplength determination < 1
72: double beta_inf; // Steplength determination < 1
73: double sigma; // Acceptance criteria < 1)
74: double minimumStep; // Minimum step size
75: double lastReference; // Reference value of last iteration
77: int memorySize; // Number of functions kept in memory
78: int current; // Current element for FIFO
79: int referencePolicy; // Integer for reference calculation rule
80: int replacementPolicy; // Policy for replacing values in memory
81: } TAO_ARMIJO;
83: #endif