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A new type of stepsize, which was recently introduced by Liu and Liu(Optimization, 67(3), 427-440, 2018), is called approximately optimal stepsizeand is quit efficient for gradient method. Interestingly, all gradient methodscan be regarded as gradient methods with approximately optimal stepsizes. Inthis paper, based on the work (Numer. Algorithms 78(1), 21-39, 2018), wepresent an improved gradient method with approximately optimal stepsize basedon conic model for unconstrained optimization. If the objective function $ f $is not close to a quadratic on the line segment between the current and latestiterates, we construct a conic model to generate approximately optimal stepsizefor gradient method if the conic model can be used; otherwise, we constructsome quadratic models to generate approximately optimal stepsizes for gradientmethod. The convergence of the proposed method is analyzed under suitableconditions. Numerical comparisons with some well-known conjugate gradientsoftware packages such as CG$ \_ $DESCENT (SIAM J. Optim. 16(1), 170-192, 2005)and CGOPT (SIAM J. Optim. 23(1), 296-320, 2013) indicate the proposed method isvery promising.

An Improved Gradient Method with Approximately Optimal Stepsize Based on Conic model for Unconstrained Optimization