Solving optimization problems by searching along a curve

Abstract

This paper proposes a curve for the one‐dimensional search method in optimization problems. The search is along a curve instead of a line. This curve is determined by a parameter [u] from the following three points: (1) the initial point X_o, (2) the Cauchy point X_c, (3) the Newton point X_N, and has the following four characteristics: (1) tangents to the steepest descent direction at X_o, (2) passes through X_N, (3) decreases monotonically from X_o to X_N for a quadratic function, (4) no complex computation on the parameter u. Therefore, the proposed method has the following advantages: (1) it is globally convergent, (2) it is locally q‐quadratically (or q‐superlinearly for quasi‐Newton point) convergent, (3) the search procedure is as simple as the line search method.