The optimal mean-reverting portfolio (MRP) design problem is an important task for statistical arbitrage, also known as pairs trading, in the financial markets. The target of the problem is to construct a portfolio of the underlying assets (possibly with an asset selection target) that can exhibit a satisfactory mean reversion property and a desirable variance property. In this paper, the optimal MRP design problem is studied under an investment leverage constraint representing the total investment positions on the underlying assets. A general problem formulation is proposed by considering the design targets subject to a leverage constraint. To solve the problem, a unified optimization framework based on the successive convex approximation method is developed. The superior performance of the proposed formulation and the algorithms are verified through numerical simulations on both synthetic data and real market data.