<p>An efficient approximation-based optimization framework is proposed for concurrent topology and mixed-variable sizing (discrete- and continuous-valued) problems. Unlike conventional metaheuristics that directly explore the combinatorial space at high computational cost, the method constructs a series of explicit subproblems using an adjusted branched multi-point approximation technique. Move limits for topology and both types of size variables are adaptively controlled to ensure stability. An improved genetic algorithm with hybrid encoding and multi-level operators is designed to solve each approximate subproblem. The initial population is generated based on the concept of adjacent individuals,and tailored crossover and mutation operators subsequently maintain the proximity of solutions to prior optima,balancing approximation stability and population diversity. The strategy is verified through benchmark cases including truss structures, rib-stiffened shells, composite laminates, and sandwich panels and further applied to satellite structural optimization. The results demonstrate that the framework achieves competitive solution quality with only several to dozens of structural and sensitivity analyses required, which is far fewer than direct metaheuristic searches, thus significantly improving optimization efficiency for complex engineering structures.</p>