<p>The concave-convex saddle point problems have a variety of applications in image processing, artificial intelligence, machine learning and other fields. In this paper, we combine the strategies of extrapolations acceleration and linesearch to give a double extrapolations primal-dual algorithm with linesearch for solving this problems. Then, the global convergence are witnessed and we also establish the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O\left( \frac{1}{T} \right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation> ergodic convergence rate in the general convex case. Moreover, when a module of objective function is strongly convex, our algorithm shows a accelerated <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(O\left( \frac{1}{T^{2}} \right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <msup> <mi>T</mi> <mn>2</mn> </msup> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation> convergence rate via the adaptive optimization of step ratio parameter, where <i>T</i> is the maximum number of iterations. Finally, numerical experiments on the ROF model of image denoising verify the effectiveness of the proposed algorithms. The numerical results indicate the algorithms proposed require less computing time and iterations than some existing primal-dual algorithms.</p>