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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