Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

/pdf/mixed-integer-nonlinear-optimization-3yf57zxyv3.pdf

Mixed-integer nonlinear optimization

Linear active-power-only power flow approximations are pervasive in the planning and control of power systems However, AC power systems are governed by a system of nonlinear nonconvex power flow equations Existing linear approximations fail to capture key power flow variables, including reactive power and voltage magnitudes, both of which are necessary in many applications that require voltage management and AC power flow feasibility This paper proposes novel linear-programming models (the LPAC models) that incorporate reactive power and voltage magnitudes in a linear power flow approximation The LPAC models are built on a polyhedral relaxation of the cosine terms in the AC equations as well as Taylor approximations of the remaining nonlinear terms Experimental comparisons with AC solutions on a variety of standard IEEE and Matpower benchmarks show that the LPAC models produce accurate values for active and reactive power, phase angles, and voltage magnitudes The potential benefits of the LPAC model

A Linear-Programming Approximation of AC Power Flows

A survey of adjustable robust optimization

Linear active-power-only DC power flow approximations are pervasive in the planning and control of power systems. However, these approximations fail to capture reactive power and voltage magnitudes, both of which are necessary in many applications to ensure voltage stability and AC power flow feasibility. This paper proposes linear-programming models (the LPAC models) that incorporate reactive power and voltage magnitudes in a linear power flow approximation. The LPAC models are built on a convex approximation of the cosine terms in the AC equations, as well as Taylor approximations of the remaining nonlinear terms. Experimental comparisons with AC solutions on a variety of standard IEEE and MatPower benchmarks show that the LPAC models produce accurate values for active and reactive power, phase angles, and voltage magnitudes. The potential benefits of the LPAC models are illustrated on two "proof-of-concept" studies in power restoration and capacitor placement.

http://www.sarahgnurre.com/Research_files/EJOR2012.pdf

Restoring infrastructure systems: An integrated network design and scheduling (INDS) problem

Using mixed-integer programming to solve power grid blackout problems

Metric inequalities and the Network Loading Problem

In this paper the Robust Network Loading problem with splittable flows and dynamic routing under polyhedral uncertainty for the demands is considered. Polyhedral results for the capacity formulation of the problem are given. The first exact approach for solving the problem is presented. A branch-and-cut algorithm based on the proposed capacity formulation is developed. Computational results using the hose polyhedron to model the demand uncertainty are discussed.

/pdf/the-robust-network-loading-problem-with-dynamic-routing-loib1pydo7.pdf

The robust network loading problem with dynamic routing

http://eprints.bice.rm.cnr.it/1445/1/tr_6_2010.pdf

Bounded coloring of co-comparability graphs and the pickup and delivery tour combination problem

We address the problem of designing a multi-layer network with survivability requirements. We are given a two-layer network: the lower layer represents the potential physical connections that can be activated, the upper layer is made of logical connections that can be set up using physical links. We are given origin-destination demands (commodities) to be routed at the upper layer. We are also given a set of failure scenarios and, for every scenario, an associated subset of commodities. The goal is to install minimum cost integer capacities on the links of both layers in order to ensure that the commodities can be routed simultaneously on the network. In addition, in every failure scenario the routing of the associated commodities must be guaranteed. We consider two variants of the problem and develop a branch-and-cut scheme based on the capacity formulation. Computational results on instances derived from the SNDLib for single node failure scenarios are discussed.

/pdf/solving-survivable-two-layer-network-design-problems-by-39d9u6fyv5.pdf

Sara Mattia

Papers

Using mixed-integer programming to solve power grid blackout problems

Metric inequalities and the Network Loading Problem

The robust network loading problem with dynamic routing

Bounded coloring of co-comparability graphs and the pickup and delivery tour combination problem

Solving survivable two-layer network design problems by metric inequalities