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We propose new sequential simulation-optimization algorithms for general convex optimization via simulation problems with high-dimensional discrete decision space. The performance of each choice of discrete decision variables is evaluated via stochastic simulation replications. If an upper bound on the overall level of uncertainties is known, our proposed simulation-optimization algorithms utilize the discrete convex structure and are guaranteed with high probability to find a solution that is close to the best within any given user-specified precision level. The proposed algorithms work for any general convex problem and the efficiency is demonstrated by proven upper bounds on simulation costs. The upper bounds demonstrate a polynomial dependence on the dimension and scale of the decision space. For some discrete optimization via simulation problems, a gradient estimator may be available at low costs along with a single simulation replication. By integrating gradient estimators, which are possibly biased, we propose simulation-optimization algorithms to achieve optimality guarantees with a reduced dependence on the dimension under moderate assumptions on the bias.