## 90C20 Quadratic programming

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#### Keywords

- Large Neighborhood Search (2)
- Mixed-Integer Nonlinear Programming (2)
- Primal Heuristic (1)
- Mixed-Integer Quadratically Constrained Programming (1)
- Mixed-Integer Quadratically Constrained Programming (1)
- Mixed-Intger Programming (1)
- Nonconvex Optimization (1)
- Primal Heuristic (1)
- bisection (1)
- constraint integer programming (1)

#### Institute

- RENS – the optimal rounding (2012)
- This article introduces RENS, the relaxation enforced neighborhood search, a large neighborhood search algorithm for mixed integer nonlinear programming (MINLP) that uses a sub-MINLP to explore the set of feasible roundings of an optimal solution x' of a linear or nonlinear relaxation. The sub-MINLP is constructed by fixing integer variables x_j with x'_j in Z and bounding the remaining integer variables to x_j in {floor(x'_j), ceil(x'_j)}. We describe two different applications of RENS: as a standalone algorithm to compute an optimal rounding of the given starting solution and as a primal heuristic inside a complete MINLP solver. We use the former to compare different kinds of relaxations and the impact of cutting planes on the roundability of the corresponding optimal solutions. We further utilize RENS to analyze the performance of three rounding heuristics implemented in the branch-cut-and-price framework SCIP. Finally, we study the impact of RENS when it is applied as a primal heuristic inside SCIP. All experiments were performed on three publically available test sets of mixed integer linear programs (MIPs), mixed integer quadratically constrained programs (MIQCPs), and MINLPs, using solely software which is available in source code. It turns out that for these problem classes 60% to 70% of the instances have roundable relaxation optima and that the success rate of RENS does not depend on the percentage of fractional variables. Last but not least, RENS applied as primal heuristic complements nicely with existing root node heuristics in SCIP and improves the overall performance.

- Undercover: a primal MINLP heuristic exploring a largest sub-MIP (2012)
- We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programming (MINLP) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a minimal set of variables that need to be fixed in order to linearize each constraint, a so-called cover. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. We apply domain propagation and conflict analysis to try to avoid infeasibilities and learn from them, respectively. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and general MINLPs from MINLPLib. It turns out that the majority of these instances allow for small covers. Although general in nature, the heuristic appears most promising for MIQCPs, and complements nicely with existing root node heuristics in different state-of-the-art solvers.

- Extending a CIP framework to solve MIQCPs (2009)
- This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MIP and CP technologies. In particular, this addresses the linear relaxation and the discrete components of the problem. For relaxation, we use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently. We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicates the potential of the approach.

- A Binary Quadratic Programming Approach to the Vehicle Positioning Problem (2009)
- The Vehicle Positioning Problem (VPP) consists of the assignment of vehicles (buses, trams or trains) of a public transport or railway company to parking positions in a depot and to timetabled trips. Such companies have many different types of vehicles, and each trip can be performed only by vehicles of some of these types. These assignments are non-trivial due to the topology of depots. The parking positions are organized in tracks, which work as one- or two-sided stacks or queues. If a required type of vehicle is not available in the front of any track, shunting movements must be performed in order to change vehicles' positions, which is undesirable and should be avoided. In this text we present integer linear and non-linear programming formulations for some versions of the problem and compare them from a theoretical and a computational point of view.

- A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations (2001)
- The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 0-1 programming problems by odd-cycle inequalities. We also consider separating odd-cycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for max-cut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably.

- Markowitz Revisited: Single-Period and Multi-Period Mean-Variance Models (1999)
- Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We investigate in detail the interplay between objective and constraints in a number of single-period variants, including semi-variance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multi-period models based on scenario trees. A key property is the possibility to remove surplus money in future decisions, yielding approximate downside risk minimization.