Ultrasound elastography involves imaging tissue while it undergoes deformation and inferring its mechanical properties from the deformation pattern. The initial deformation in the tissue is typically induced through an external mechanical force, for example, by exerting a slight pressure using an ultrasound probe or by applying an acoustic radiation force (ARF) against the tissue. The ARF excites the tissue locally, which leads to the propagation of a shear-wave. The goal of the shear-wave elastography is to estimate the speed of the shear-wave that is explicitly related to the elasticity of tissue. We formulate tissue displacement estimation as an optimization problem and propose a computationally efficient approach to estimate the displacement field. A novel algorithm based on the minimization of a regularized cost function using higher-order analytical minimization (HAM) coupled with the second-order Taylor series approximation is proposed. Our algorithm first computes an integer displacement field based on dynamic programming (DP) that provides the global optima, which is then refined iteratively to obtain the subpixel displacement estimate. We test the proposed algorithm on real experimental data obtained from a tissue-mimicking phantom and illustrate the superiority of our approach over some commonly used elastography techniques using signal to noise ratio (SNR) comparisons.