This study considers some algorithms for solving wide classes of convex optimization problems based on first-order information where the underlying function has a large number of variables. The primary idea is to introduce an optimal subgradient-based algorithm by proposing a novel fractional subproblem and explicitly solve it to be appropriate for employing in applications like signal and image processing, machine learning, statistics and so on. Then we develop the basic algorithm by carefully incorporating an affine term and a multidimensional subspace search into it to improve implementations of the algorithm when considered problems involve composition of an expensive linear mapping along with a computationally inexpensive function. The next aim is to introduce some prox function for unconstrained version of problems and effectively solve the corresponding subproblem. We also prove that the proposed algorithms are optimal in the sense of complexity for both smooth and nonsmooth functions. Considering an unconstrained version of some highly practical problems in signal and image processing, machine learning and statistics, we report some numerical results and compare with some state-of-the-art solvers.