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The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and secondorder cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with a predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems.