For the fast ignition scheme of laser fusion, a highly compressed fuel core is necessary using a cone-inserted spherical solid target. We design an optimal compression method with a multistep laser pulse shape using a 2D radiation hydrodynamics simulation. In the design, the maximum areal density is ρR max = 0.46 g cm-2 with 8 kJ of implosion laser. Considering the scaling law of hydrodynamic, we obtained 82 kJ and 1.3 MJ for the ignition-scale-target (ρR max = 1.1 g cm-2) and burning-scale-target (ρR max = 2.5 g cm-2), respectively. In a preliminary study of hydrodynamic instability in the optimized cone-inserted implosion, there is no significant growth of the instabilities that affect the implosion performance. A parameter search to investigate the feasibility and robustness of a high areal density design for changes in the laser pulse shape is performed. As a result, a delay of several tens of pico-second of the rising laser pulse is acceptable.