A linear programming based algorithm to solve a class of optimization problems with a multi-linear objective function and affine constraints

- We study a class of convex optimization problems with a multi-linear objective.
- We develop a novel linear programming based algorithm.
- We show that the proposed algorithm outperforms standard solvers.

We present a linear programming based algorithm for a class of optimization problems with a multi-linear objective function and affine constraints. This class of optimization problems has only one objective function, but it can also be viewed as a class of multi-objective optimization problems by decomposing its objective function. The proposed algorithm exploits this idea and solves this class of optimization problems from the viewpoint of multi-objective optimization. The algorithm computes an optimal solution when the number of variables in the multi-linear objective function is two, and an approximate solution when the number of variables is greater than two. A computational study demonstrates that when available computing time is limited the algorithm significantly outperforms well-known convex programming solvers IPOPT and CVXOPT, in terms of both efficiency and solution quality. The optimization problems in this class can be reformulated as second-order cone programs, and, therefore, also be solved by second-order cone programming solvers. This is highly effective for small and medium size instances, but we demonstrate that for large size instances with two variables in the multi-linear objective function the proposed algorithm outperforms a (commercial) second-order cone programming solver.