## 90C26 Nonconvex programming, global optimization

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- ZIB-Report (17)
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#### Keywords

- Pooling Problem (4)
- nonconvex (3)
- Cutting Planes (2)
- Large Neighborhood Search (2)
- MINLP (2)
- Nonconvexity (2)
- Primal Heuristic (2)
- Quadratic Programming (2)
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#### Institute

- Mathematical Optimization (19) (remove)

This paper describes the extensions that were added to the constraint integer programming framework SCIP in order to enable it to solve convex and nonconvex mixed-integer nonlinear programs (MINLPs) to global optimality.
SCIP implements a spatial branch-and-bound algorithm based on a linear outer-approximation, which is computed by convex over- and underestimation of nonconvex functions. An expression graph representation of nonlinear constraints allows for bound tightening, structure analysis, and reformulation. Primal heuristics are employed throughout the solving process to find feasible solutions early. We provide insights into the performance impact of individual MINLP solver components via a detailed computational study over a large and heterogeneous test set.

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.

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.

The SCIP Optimization Suite is a powerful collection of optimization software that consists of the branch-cut-and-price framework and mixed-integer programming solver SCIP, the linear programming solver SoPlex, the modeling language Zimpl, the parallelization framework UG, and the generic branch-cut-and-price solver GCG. Additionally, it features the extensions SCIP-Jack for solving Steiner tree problems, PolySCIP for solving multi-objective problems, and SCIP-SDP for solving mixed-integer semidefinite programs. The SCIP Optimization Suite has been continuously developed and has now reached version 4.0. The goal of this report is to present the recent changes to the collection. We not only describe the theoretical basis, but focus on implementation aspects and their computational consequences.

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.

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.

Optimization-based bound tightening (OBBT) is a domain reduction technique commonly used in nonconvex mixed-integer nonlinear programming that solves a sequence of auxiliary linear programs. Each variable is minimized and maximized to obtain the tightest bounds valid for a global linear relaxation. This paper shows how the dual solutions of the auxiliary linear programs can be used to learn what we call Lagrangian variable bound constraints. These are linear inequalities that explain OBBT's domain reductions in terms of the bounds on other variables and the objective value of the incumbent solution. Within a spatial branch-and-bound algorithm, they can be learnt a priori (during OBBT at the root node) and propagated within the search tree at very low computational cost. Experiments with an implementation inside the MINLP solver SCIP show that this reduces the number of branch-and-bound nodes and speeds up solution times.

This paper is concerned with optimal operation of pressurized water supply networks at a fixed point in time. We use a mixed-integer nonlinear programming (MINLP) model incorporating both the nonlinear physical laws and the discrete decisions such as switching pumps on and off. We demonstrate that for instances from our industry partner, these stationary models can be solved to ε-global optimality within small running times using problem-specific presolving and state-of-the-art MINLP algorithms.
In our modeling, we emphasize the importance of distinguishing between what we call real and imaginary flow, i.e., taking into account that the law of Darcy-Weisbach correlates pressure difference and flow along a pipe if and only if water is available at the high pressure end of a pipe. Our modeling solution extends to the dynamic operative planning problem.

In this paper we investigate the performance of several out-of-the box solvers for mixed-integer quadratically constrained programmes (MIQCPs) on an open pit mine production scheduling problem with mixing constraints. We compare the solvers BARON, Couenne, SBB, and SCIP to a problem-specific algorithm on two different MIQCP formulations. The computational results presented show that general-purpose solvers with no particular knowledge of problem structure are able to nearly match the performance of a hand-crafted algorithm.

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.