In a generic wireless network where the activation of a transmission link is subject to its signal-to-noise-and-interference ratio (SINR) constraint, one of the most fundamental and yet challenging problem is to find the maximum number of simultaneous transmissions. In this thesis, we consider and study in detail the problem of maximum link activation in wireless networks based on the SINR model. Integer Linear Programming has been used as the main tool in this thesis for the design of algorithms. Fast algorithms have been proposed for the delivery of near-optimal results time-efficiently. With the state-of-art Gurobi optimization solver, both the conventional approach consisting of all the SINR constraints explicitly and the exact algorithm developed recently using cutting planes have been implemented in the thesis. Based on those implementations, new solution algorithms have been proposed for the fast delivery of solutions. Instead of considering interference from all other links, an interference range has been proposed. Two scenarios have been considered, namely the optimistic case and the pessimistic case. The optimistic case considers no interference from outside the interference range, while the pessimistic case considers the interference from outside the range as a common large value. Together with the algorithms, further enhancement procedures on the data analysis have also been proposed to facilitate the computation in the solver.

This paper explores strategies in portfolio optimization, focusing on integrating mean-variance optimization (MVO) frameworks with cardinality constraints to enhance investment decision-making. Using a combination of quadratic programming and mixed-integer linear programming, the Gurobi optimizer handles complex constraints and achieves computational solutions. The study compares two mathematical formulations of the cardinality constraint: the Complementary Model and the Big M Model. As cardinality increased, risk decreased exponentially, converging at higher cardinalities. This behavior aligns with the theory of risk reduction through diversification. Additionally, despite initial expectations, both models performed similarly in terms of root relaxation risk and execution time due to Gurobi's presolve transformation of the Complementary Model into the Big M Model. Root relaxation risks were identical while execution times varied slightly without a consistent trend, underscoring the Big M Model's versatility and highlighting the limitations of the Complementary Model.