Hoa Thi Bui

Cutting plane methods, particularly outer approximation, are a well-established approach for solving nonlinear discrete optimization problems without relaxing the integrality of decision variables. While powerful in theory, their computational performance can be highly variable. Recent research has shown that constructing cutting planes at the projection of infeasible points onto the feasible set can significantly improve the performance of cutting plane approaches. Motivated by this, we examine whether constructing cuts at feasible points closer to the optimal solution set could further enhance the effectiveness of cutting plane methods. We propose a hybrid method that combines the global convergence guarantees of cutting plane methods with the local exploration capabilities of first-order optimization techniques. Specifically, we use projected gradient methods as a heuristic to identify promising regions of the solution space and generate tighter, more informative cuts. We focus on binary optimization problems with convex differentiable objective functions, where projection operations can be efficiently computed via mixed-integer linear programming. By constructing cuts at points closer to the optimal solution set and eliminating non-optimal regions, the algorithm achieves better approximation of the feasible region and faster convergence. Numerical experiments confirm that our approach improves both the quality of the solution and computational efficiency across different solver configurations. This framework provides a flexible foundation for further extensions to more general discrete domains and offers a promising heuristic to the toolkit for nonlinear discrete optimization.

Divide and Conquer: Solving the Euclidean Diversity Problem Through Coordinate Partitioning A common technique for solving difficult optimization problems is to break them into easier-to-manage pieces, yet it is rarely obvious how to do so. This is especially true for the Euclidean max-sum diversity problem, which becomes increasingly difficult as the number of coordinates grows despite the problem size remaining the same. In “Coordinate Partitioning for Difficult Euclidean Max-Sum Diversity Problems,” Spiers, Bui, and Loxton use an old result by Schoenberg to transform Euclidean distances into their squared counterparts, enabling a functional decomposition of the problem’s objective. The authors introduce novel partitioning strategies as well as a globally convergent cutting-plane algorithm, which can effectively solve high-dimension instances. They also present a new class of challenging instances characterized by locations on the surface of a hyperball.

The cutting plane method is known to converge for concave discrete maximization problems. This paper introduces a new condition for convergence that is weaker than concavity, allowing for extensions to generalized concave cases such as quasiconcavity and pseudoconcavity. More significantly, the new convergence condition enables the use of smaller penalty values in the standard concave reformulation technique for general nonconcave problems, yielding stronger cuts and faster convergence. In concave cases, this condition can even be applied to “deconcavify” concave problems to enhance convergence speed. Numerical results show that exploiting the new convergence condition can yield substantial computational improvements for complex, large-scale discrete optimization problems with up to one thousand variables.

We consider a class of convex optimization problems in a Hilbert space that can be solved by performing a single projection, i.e., by projecting an infeasible point onto the feasible set. Our results improve those established for the linear programming setting in Nurminski (2015) by considering problems that (i) may have multiple solutions, (ii) do not satisfy strict complementarity conditions, and (iii) possess nonlinear convex constraints. As a by-product of our analysis, we provide a quantitative estimate on the required distance between the infeasible point and the feasible set in order for its projection to be a solution of the problem. Our analysis relies on a “sharpness” property of the constraint set, a new property we introduce here.

In this work, we consider a class of convex optimization problems in a real Hilbert space that can be solved by performing a single projection, i.e., by projecting an infeasible point onto the feasible set. Our results improve those established for the linear programming setting in Nurminski (2015) by considering problems that: (i) may have multiple solutions, (ii) do not satisfy strict complementary conditions, and (iii) possess non-linear convex constraints. As a by-product of our analysis, we provide a quantitative estimate on the required distance between the infeasible point and the feasible set in order for its projection to be a solution of the problem. Our analysis relies on a "sharpness" property of the constraint set; a new property we introduce here.

Maintenance activities are inevitable and costly in integrated mining operations. Conducting maintenance may require the whole system, or sub-units of the system, to be shut down temporarily. These maintenance activities not only disrupt the unit being shut down, but they also have consequences for inventory levels and product flow downstream. In this paper, we consider an interconnected mining system in which there are complicated maintenance relationships and stock accumulation at intermediate nodes. We propose a time-indexed mixed-integer linear programming formulation to optimize the long-term integrated maintenance plan and maximize the total throughput. We also devise an algorithm, which combines Benders decomposition and Lagrangian relaxation, to accelerate the computational speed. To validate our mathematical model, we perform simulations for a real-world case study in the iron ore industry. The results show that our method can yield better solutions than optimization solver alone in faster time.