## 90C20 Quadratic programming

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#### Document Type

- ZIB-Report (9)
- Doctoral Thesis (1)

#### Keywords

- Pooling Problem (4)
- Cutting Planes (2)
- Large Neighborhood Search (2)
- Mixed-Integer Nonlinear Programming (2)
- Nonconvexity (2)
- Primal Heuristic (2)
- Quadratic Programming (2)
- Relaxation (2)
- Relaxations (2)
- Standard Quadratic Programming (2)

#### Institute

The amazing success of computational mathematical optimization over the last decades has been driven more by insights into mathematical structures than by the advance of computing technology. In this vein, we address applications, where nonconvexity in the model poses principal difficulties.
This paper summarizes the dissertation of Jonas Schweiger for the occasion of the GOR dissertation award 2018. We focus on the work on non-convex quadratic programs and show how problem specific structure can be used to obtain tight relaxations and speed up Branch&Bound methods. Both a classic general QP and the Pooling Problem as an important practical application serve as showcases.

We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which products are mixed in intermediate pools in order to meet quality targets at their destinations. In this technical report, we characterize the extreme points of the convex hull of our non-convex set, and show that they are not finite, i.e., the convex hull is not polyhedral. This analysis was used to derive valid nonlinear convex inequalities and show that, for a specific case, they characterize the convex hull of our set. The new valid inequalities and computational results are presented in ZIB Report 18-12.

We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which input materials are mixed in intermediate pools, with the outputs of these pools further mixed to make output products meeting given attribute percentage requirements. Our relaxations are derived by considering a set which arises from the formulation by considering a single product, a single attibute, and a single pool. The convex hull of the resulting nonconvex set is not polyhedral. We derive valid linear and convex nonlinear inequalities for the convex hull, and demonstrate that different subsets of these inequalities define the convex hull of the nonconvex set in three cases determined by the parameters of the set. Computational results on literature instances and newly created larger test instances demonstrate that the inequalities can significantly strengthen the convex relaxation of the pq-formulation of the pooling problem, which is the relaxation known to have the strongest bound.

Exploiting structure in non-convex quadratic optimization and gas network planning under uncertainty
(2017)

The amazing success of computational mathematical optimization over
the last decades has been driven more by insights into mathematical
structures than by the advance of computing technology. In this vein,
we address applications, where nonconvexity in the model and
uncertainty in the data pose principal difficulties.
The first part of the thesis deals with non-convex quadratic programs.
Branch&Bound methods for this problem class depend on tight
relaxations. We contribute in several ways: First, we establish a new
way to handle missing linearization variables in the well-known
Reformulation-Linearization-Technique (RLT). This is implemented
into the commercial software CPLEX. Second, we study the optimization
of a quadratic objective over the standard simplex or a knapsack
constraint. These basic structures appear as part of many complex
models. Exploiting connections to the maximum clique problem and RLT,
we derive new valid inequalities. Using exact and heuristic separation
methods, we demonstrate the impact of the new inequalities on the
relaxation and the global optimization of these problems. Third, we
strengthen the state-of-the-art relaxation for the pooling problem, a
well-known non-convex quadratic problem, which is, for example,
relevant in the petrochemical industry. We propose a novel relaxation
that captures the essential non-convex structure of the problem but is
small enough for an in-depth study. We provide a complete inner
description in terms of the extreme points as well as an outer
description in terms of inequalities defining its convex hull (which
is not a polyhedron). We show that the resulting valid convex
inequalities significantly strengthen the standard relaxation of the
pooling problem.
The second part of this thesis focuses on a common challenge in real
world applications, namely, the uncertainty entailed in the input
data.
We study the extension of a gas transport network, e.g., from our
project partner Open Grid Europe GmbH.
For a single scenario this maps to a challenging non-convex MINLP.
As the future transport patterns are highly uncertain, we propose a
robust model to best prepare the network operator for an array of
scenarios.
We develop a custom decomposition approach that makes use of the
hierarchical structure of network extensions and the loose coupling
between the scenarios.
The algorithm used the single-scenario problem as black-box subproblem
allowing the generalization of our approach to problems with the same
structure.
The scenario-expanded version of this problem is out of reach for
today's general-purpose MINLP solvers.
Yet our approach provides primal and dual bounds for instances with up
to 256 scenarios and solves many of them to optimality.
Extensive computational studies show the impact of our work.

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.

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.

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.

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.

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.

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.