### Abstract

The conjugate point is an important global concept in the calculus of variations and optimal control. In these extremal problems, the variable is not a vector in R^{n} but a function. So a simple and natural question arises. Is it possible to establish a conjugate points theory for a nonlinear programming problem, Min f(x) on x ∈ R^{n}? This paper positively answers this question. We introduce the Jacobi equation and conjugate points for the nonlinear programming problem, and we describe necessary and sufficient optimality conditions in terms of conjugate points.

Original language | English |
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Pages (from-to) | 54-63 |

Number of pages | 10 |

Journal | SIAM Journal on Control and Optimization |

Volume | 40 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics
- Control and Optimization

### Cite this

**A conjugate points theory for a nonlinear programming problem.** / Kawasaki, Hidefumi.

Research output: Contribution to journal › Article

*SIAM Journal on Control and Optimization*, vol. 40, no. 1, pp. 54-63. https://doi.org/10.1137/S0363012900368831

}

TY - JOUR

T1 - A conjugate points theory for a nonlinear programming problem

AU - Kawasaki, Hidefumi

PY - 2002

Y1 - 2002

N2 - The conjugate point is an important global concept in the calculus of variations and optimal control. In these extremal problems, the variable is not a vector in Rn but a function. So a simple and natural question arises. Is it possible to establish a conjugate points theory for a nonlinear programming problem, Min f(x) on x ∈ Rn? This paper positively answers this question. We introduce the Jacobi equation and conjugate points for the nonlinear programming problem, and we describe necessary and sufficient optimality conditions in terms of conjugate points.

AB - The conjugate point is an important global concept in the calculus of variations and optimal control. In these extremal problems, the variable is not a vector in Rn but a function. So a simple and natural question arises. Is it possible to establish a conjugate points theory for a nonlinear programming problem, Min f(x) on x ∈ Rn? This paper positively answers this question. We introduce the Jacobi equation and conjugate points for the nonlinear programming problem, and we describe necessary and sufficient optimality conditions in terms of conjugate points.

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UR - http://www.scopus.com/inward/citedby.url?scp=0036203751&partnerID=8YFLogxK

U2 - 10.1137/S0363012900368831

DO - 10.1137/S0363012900368831

M3 - Article

VL - 40

SP - 54

EP - 63

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

SN - 0363-0129

IS - 1

ER -