Minimizing conductor cost is one of the goals in high-field HTS magnet design. Optimization of magnet design to minimize conductor length is one way to approach the problem. HTS tapes have anisotropic material properties and non-uniform current density within the tape width during operation, which must be accounted in modeling HTS tapes. Finite element method (FEM) based simulations can be used to obtain accurate magnetic field distribution and critical current in an HTS magnet, but these simulations are typically time consuming. For example, a detailed simulation for a 2 T HTS solenoid may take several hours. As a result, it is not efficient to implement the traditional optimization algorithms, which directly use the simulation results to obtain the response of the cost function. To overcome this challenge, a novel optimization algorithm was developed by the authors, namely L-ANN-GWO to reduce the time cost of optimization (L stands for LASSO, ANN stands for artificial neural network, and GWO stands for grey wolf optimizer). In this approach, ANN is trained by FEA set of HTS solenoid designs to approximate the time-consuming simulations of magnetic field distribution based on solenoid geometry. Instead of using the approximation model in a static way, ANN is first trained with a small number of samples and updated adaptively in L-ANN-GWO along with the optimization process. In this contribution, we present application of the L-ANN-GWO optimization algorithm to ReBCO solenoid coils to optimize magnet design to minimize conductor use. The design constraints come from field homogeneity and critical surface. We demonstrate that as a multi-objective optimization, L-ANN-GWO can output the minimized superconductor consumption in different peak magnetic fields from a single optimization run. Future developments foresee adding quench protection requirements into the design optimization as this is another aspect potentially impacting the conductor use.