### Abstract

The conjugate point is a global concept in the calculus of variations. It plays an important role in second-order optimality conditions. A conjugate point theory for a minimization problem of a smooth function with n variables was proposed in (H. Kawasaki (2000). Conjugate points for a nonlinear programming problem with constraints. J. Nonlinear Convex Anal., 1,287-293; H. Kawasaki (2001). A conjugate points theory for a nonlinear programming problem. SIAM J. Control Optim., 40, 54-63.). In those papers, we defined the Jacobi equation and (strict) conjugate points, and derived necessary and sufficient optimality conditions in terms of conjugate points. The aim of this article is to analyze conjugate points for tridiagonal Hesse matrices of a class of extremal problems. We present a variety of examples, which can be regarded as a finite-dimensional analogy to the classical shortest path problem on a surface.

Original language | English |
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Pages (from-to) | 197-205 |

Number of pages | 9 |

Journal | Optimization Methods and Software |

Volume | 18 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Software
- Control and Optimization
- Applied Mathematics

### Cite this

**Analysis of conjugate points for constant tridiagonal Hesse matrices of a class of extremal problems.** / Kawasaki, Hidefumi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Analysis of conjugate points for constant tridiagonal Hesse matrices of a class of extremal problems

AU - Kawasaki, Hidefumi

PY - 2003/4/1

Y1 - 2003/4/1

N2 - The conjugate point is a global concept in the calculus of variations. It plays an important role in second-order optimality conditions. A conjugate point theory for a minimization problem of a smooth function with n variables was proposed in (H. Kawasaki (2000). Conjugate points for a nonlinear programming problem with constraints. J. Nonlinear Convex Anal., 1,287-293; H. Kawasaki (2001). A conjugate points theory for a nonlinear programming problem. SIAM J. Control Optim., 40, 54-63.). In those papers, we defined the Jacobi equation and (strict) conjugate points, and derived necessary and sufficient optimality conditions in terms of conjugate points. The aim of this article is to analyze conjugate points for tridiagonal Hesse matrices of a class of extremal problems. We present a variety of examples, which can be regarded as a finite-dimensional analogy to the classical shortest path problem on a surface.

AB - The conjugate point is a global concept in the calculus of variations. It plays an important role in second-order optimality conditions. A conjugate point theory for a minimization problem of a smooth function with n variables was proposed in (H. Kawasaki (2000). Conjugate points for a nonlinear programming problem with constraints. J. Nonlinear Convex Anal., 1,287-293; H. Kawasaki (2001). A conjugate points theory for a nonlinear programming problem. SIAM J. Control Optim., 40, 54-63.). In those papers, we defined the Jacobi equation and (strict) conjugate points, and derived necessary and sufficient optimality conditions in terms of conjugate points. The aim of this article is to analyze conjugate points for tridiagonal Hesse matrices of a class of extremal problems. We present a variety of examples, which can be regarded as a finite-dimensional analogy to the classical shortest path problem on a surface.

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UR - http://www.scopus.com/inward/citedby.url?scp=0038487307&partnerID=8YFLogxK

U2 - 10.1080/1055678031000109554

DO - 10.1080/1055678031000109554

M3 - Article

AN - SCOPUS:0038487307

VL - 18

SP - 197

EP - 205

JO - Optimization Methods and Software

JF - Optimization Methods and Software

SN - 1055-6788

IS - 2

ER -