A parameter-free optimization technique is applied in Quasi-Newton’s method for solving unconstrained multiobjective optimization problems. The components of the Hessian matrix are constructed using q-derivative, which is positive definite at every iteration. The step-length is computed by an Armijo-like rule which is responsible to escape the point from local minimum to global minimum at every iteration due to q-derivative. Further, the rate of convergence is proved as a superlinear in a local neighborhood of a minimum point based on q-derivative. Finally, the numerical experiments show better performance.

This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation ${}^{c}\mathcal{D}_{q}^{\sigma }[k](t) = w (t, k(t), {}^{c} \mathcal{D}_{q}^{\zeta }[k](t) )$ D q σ c [ k ] ( t ) = w ( t , k ( t ) , c D q ζ [ k ] ( t ) ) with three-point conditions for $t \in (0,1)$ t ∈ ( 0 , 1 ) on a time scale $\mathbb{T}_{t_{0}}= \{ t : t =t_{0}q^{n}\}\cup \{0\}$ T t 0 = { t : t = t 0 q n } ∪ { 0 } , where $n\in \mathbb{N}$ n ∈ N , $t_{0} \in \mathbb{R}$ t 0 ∈ R , and $0< q<1$ 0 < q < 1 , based on the Leray–Schauder nonlinear alternative and Guo–Krasnoselskii theorem. Moreover, we discuss the existence of nonnegative solutions. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

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A Polak–Ribière–Polyak (PRP) algorithm is one of the oldest and popular conjugate gradient algorithms for solving nonlinear unconstrained optimization problems. In this paper, we present a q-variant of the PRP (q-PRP) method for which both the sufficient and conjugacy conditions are satisfied at every iteration. The proposed method is convergent globally with standard Wolfe conditions and strong Wolfe conditions. The numerical results show that the proposed method is promising for a set of given test problems with different starting points. Moreover, the method reduces to the classical PRP method as the parameter q approaches 1.