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As an important problem in modern data analytics, classification has witnessed varieties of applications from different domains. Different from conventional classification approaches, fair classification concerns the issues of unintentional biases against the sensitive features (e.g., gender, race). Due to high nonconvexity of fairness measures, existing methods are often unable to model exact fairness, which can cause inferior fair classification outcomes. This paper fills the gap by developing a novel unified framework to jointly optimize accuracy and fairness. The proposed framework is versatile and can incorporate different fairness measures studied in literature precisely as well as can be applicable to many classifiers including deep classification models. Specifically, in this paper, we first prove Fisher consistency of the proposed framework. We then show that many classification models within this framework can be recast as mixed-integer convex programs, which can be solved effectively by off-the-shelf solvers when the instance sizes are moderate and can be used as benchmarks to compare the efficiency of approximation algorithms. We prove that in the proposed framework, when the classification outcomes are known, the resulting problem, termed "unbiased subdata selection," is strongly polynomial-solvable and can be used to enhance the classification fairness by selecting more representative data points. This motivates us to develop an iterative refining strategy (IRS) to solve the large-scale instances, where we improve the classification accuracy and conduct the unbiased subdata selection in an alternating fashion. We study the convergence property of IRS and derive its approximation bound. More broadly, this framework can be leveraged to improve classification models with unbalanced data by taking F1 score into consideration.