| \begin{array}{rl}{0}&{\geq\left(A_{k+1}-A_{k}\right)\left[f\left(x_{k+1}\right)-f\left(x^{*}\right)+\left\langle\nabla f\left(x_{k+1}\right),x^{*}-x_{k+1}\right\rangle+\mu D_{h}\left(x^{*},x_{k+1}\right)\right]}\\&{\quad+A_{k}\left[f\left(x_{k+1}\right)-f\left(x_{k}\right)+\left\langle\nabla f\left(x_{k+1}\right),x_{k}-x_{k+1}\right\rangle\right]}\\&{\quad+A_{k+1}\left[Ms^{\frac{1}{p-1}}\left\Vert\nabla f\left(x_{k+1}\right)\right\Vert^{\frac{p}{p-1}}-\left\langle\nabla f\left(x_{k+1}\right),y_{k}-x_{k+1}\right\rangle\right]}\\&{\quad+\left(1+\mu A_{k}\right)\left[h\left(z_{k}\right)-h\left(z_{k+1}\right)+\left\langle\nabla h\left(z_{k}\right),z_{k+1}-z_{k}\right\rangle+\frac{1}{p}\left\Vert z_{k+1}-z_{k}\right\Vert^{p}\right]}\\&{\geq\mathcal{E}_{k+1}-\mathcal{E}_{k}}\\&{\quad-\mu\left(A_{k+1}-A_{k}\right)D_{h}\left(x^{*},x_{k+1}\right)+\mu\left(A_{k+1}-A_{k}\right)\left\langle\nabla h\left(z_{k+1}\right)-\nabla h\left(x_{k+1}\right),x^{*}-z_{k+1}\right\rangle}\\&{\quad-\left(A_{k+1}-A_{k}\right)\left(f\left(x_{k+1}\right)-f\left(x^{*}\right)\right)-A_{k}\left(f\left(x_{k+1}\right)-f\left(x_{k}\right)\right)}\\&{\quad-\left(1+\mu A_{k+1}\right)\left(-h\left(z_{k+1}\right)-\left\langle\nabla h\left(z_{k+1}\right),x^{*}-z_{k+1}\right\rangle+h\left(z_{k}\right)+\left\langle\nabla h\left(z_{k}\right),x^{*}-z_{k}\right\rangle\right)}\\&{\quad+\left(A_{k+1}-A_{k}\right)\left[f\left(x_{k+1}\right)-f\left(x^{*}\right)+\left\langle\nabla f\left(x_{k+1}\right),x^{*}-x_{k+1}\right\rangle+\mu D_{h}\left(x^{*},x_{k+1}\right)\right]}\\&{\quad+A_{k}\left[f\left(x_{k+1}\right)-f\left(x_{k}\right)+\left\langle\nabla f\left(x_{k+1}\right),x_{k}-x_{k+1}\right\rangle\right]}\\&{\quad+A_{k+1}\left[Ms^{\frac{1}{p-1}}\left\Vert\nabla f\left(x_{k+1}\right)\right\Vert^{\frac{p}{p-1}}-\left\langle\nabla f\left(x_{k+1}\right),y_{k}-x_{k+1}\right\rangle\right]}\\&{\quad+\left(1+\mu A_{k}\right)\left[h\left(z_{k}\right)-h\left(z_{k+1}\right)+\left\langle\nabla h\left(z_{k}\right),z_{k+1}-z_{k}\right\rangle+\frac{1}{p}\left\Vert z_{k+1}-z_{k}\right\Vert^{p}\right]}\\&{=\mathcal{E}_{k+1}-\mathcal{E}_{k}}\\&{\quad+\left\langle\nabla f\left(x_{k+1}\right),\left(A_{k+1}-A_{k}\right)\left(x^{*}-x_{k+1}\right)+A_{k}\left(x_{k}-x_{k+1}\right)+A_{k+1}\left(x_{k+1}-y_{k}\right)\right\rangle}\\&{\quad+\left(1+\mu A_{k}\right)\left\langle\nabla h\left(z_{k+1}\right)-\nabla h\left(z_{k}\right),x^{*}-z_{k+1}\right\rangle+\frac{1+\mu A_{k}}{p}\left\Vert z_{k+1}-z_{k}\right\Vert^{p}}\\&{\quad+\mu\left(A_{k+1}-A_{k}\right)\left\langle\nabla h\left(z_{k+1}\right)-\nabla h\left(x_{k+1}\right),x^{*}-z_{k+1}\right\rangle+MA_{k+1}s^{\frac{1}{p-1}}\left\Vert\nabla f\left(x_{k+1}\right)\right\Vert^{\frac{p}{p-1}}.}\end{array} |
