In this thesis we study the $l_{p}-l_{q}$ minimization problem with $ q \\leq 2 , p gt;0$ has which received significant attention in image restoration and compressive sensing . The half-quadratic regularization method is usually a technique to solve this problem . The class of $L_{1}$-regularized optimization problems has received much attention recently because of the introduction of compressed sensing , which allows images and signals to be reconstructed from small amounts of data . Despite this recent attention , many $L_{1}$-regularized problems remain difficult to solve or require very problem-specific techniques .