Optimization problems are not only formed into a linear programming but also nonlinear programming. In real life, often decision variables restricted on integer. Hence, came the nonlinear programming. One particular form of nonlinear programming is a convex quadratic programming which form the objective function is quadratic and convex and linear constraint functions. In this research designed ...

In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MIPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MIPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is that of generat...

In this paper, we study cut generating functions for conic sets. Our first main result shows that if the conic set is bounded, then cut generating functions for integer linear programs can easily be adapted to give the integer hull of the conic integer program. Then we introduce a new class of cut generating functions which are non-decreasing with respect to second-order cone. We show that, und...

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures ...

Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to generate bounds for mixed-integer linear programming problems. Traditionally, these methods have been viewed as distinct from polyhedral methods, in which bounds are obtained by dynamically generating valid inequalities to strengthen a linear programming relaxation. ...

A linear program with linear complementarity constraints (LPCC) requires the minimization of a linear objective over a set of linear constraints together with additional linear complementarity constraints. This class has emerged as a modeling paradigm for a broad collection of problems, including bilevel programs, Stackelberg games, inverse quadratic programs, and problems involving equilibrium...

In the context of finding the largest stable set of a graph, rank inequalities prescribe that a stable set can contain, from any induced subgraph of the original graph, at most as many vertices as the stability number of the former. Although these inequalities subsume many of the valid inequalities known for the problem, their exact separation has only been investigated in few special cases obt...

We study a variant of the capacitated vehicle routing problem where the cost over each arc is defined as the product of the arc length and the weight of the vehicle when it traverses that arc. We propose two new mixed integer linear programming formulations for the problem: an arc-load formulation and a set partitioning formulation based on q-routes with additional constraints. A family of cycl...

The generalized multiple depot traveling salesmen problem (GMDTSP) is a variant of the multiple depot traveling salesmen problem (MDTSP), where each salesman starts at a distinct depot, the targets are partitioned into clusters and at least one target in each cluster is visited by some salesman. The GMDTSP is an NP-hard problem as it generalizes the MDTSP and has practical applications in desig...