A Primal–Dual Interior Point Method for Complex–Variable Optimization Problems

In this paper, we propose a primal-dual interior-point method for solving convex optimization problems with complex variables, relying on a newly defined complex-valued kernel function. We extend classical kernel functions to the complex domain by establishing appropriate differentiability and convexity properties that guarantee the well-posedness and convergence of the proposed algorithm. Our theoretical approach encompasses the formulation of penalized optimality conditions, the definition of a modified Newton direction tailored to complex parametrization, and the design of a central pathtracking algorithm featuring adaptive barrier parameter updating. A rigorous complexity analysis yields polynomial bounds depending on the problem dimension and the desired accuracy. Numerical experiments on large-scale complex-variable problems demonstrate both the effectiveness and robustness of the proposed approach. The results validate the algorithm’s dimension-independence property, with iteration counts remaining stable across substantial increases in problem size, and reveal significant computational advantages over state-of-the-art general-purpose solvers including IPOPT (Interior Point Optimizer). This work advances the theoretical foundations of interior-point methods in the complex domain and opens new perspectives for high-dimensional complex optimization.