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Fractional calculus is an efficient tool, which has the potential to improve the performance of gradient methods. However, when the first order gradient direction is generalized by fractional order gradient one, the corresponding algorithm converges to the fractional extreme point of the target function which is not equal to the real extreme point. This drawback critically hampers the application of this method. To solve such a convergence problem, the current paper analyzes the specific reasons and proposes three possible solutions. Considering the long memory characteristics of fractional derivative, short memory principle is a prior choice. Apart from the truncation of memory length, two new methods are developed to reach the convergence. The former is the truncation of the infinite series, and the latter is the modification of the constant fractional order. Finally, six illustrative examples are performed to illustrate the effectiveness and practicability of proposed methods. [ABSTRACT FROM AUTHOR]

Highlights • A new nabla Taylor formula expanded at b ≥ n is derived for a discrete function. • Two kinds of series representations of are derived for nabla discrete fractional sum/difference. • Some essential properties of nabla discrete fractional calculus are investigated. • Besides fixed initial instant case, nabla fractional calculus with fixed memory step are introduced. Abstract This paper addresses the description and analysis problems of nabla discrete fractional calculus. The series representation framework is developed first, including two Taylor series expanded at the initial instant and the current time, respectively. Under this framework, several essential properties of fractional sum/difference are presented and investigated. Notably, the short memory principle is introduced for nabla discrete fractional calculus, along with which two corresponding series are studied. [ABSTRACT FROM AUTHOR]