Abstract We propose an approach based on quadratic approximations for solving general Mixed-Integer Nonlinear Programming (MINLP) problems. Specifically, our approach entails the global approximation of the epigraphs of constraint functions by means of paraboloids, which are polynomials of degree two with univariate quadratic terms, and relies on a Lipschitz property only. These approximations are then integrated into the original problem. To this end, we introduce a novel approach to compute globally valid epigraph approximations by paraboloids via a Mixed-Integer Linear Programming (MIP) model. We emphasize the possibility of performing such approximations a-priori and providing them in form of a lookup table, and then present several ways of leveraging the approximations to tackle the original problem. We provide the necessary theoretical background and conduct computational experiments on instances of the MINLPLib. As a result, this approach significantly accelerates the solution process of MINLP problems, particularly those involving many trigonometric or few exponential functions. In general, we highlight that the proposed technique is able to exploit advances in Mixed-Integer Quadratically-Constrained Programming (MIQCP) to solve MINLP problems.

Abstract In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz constant. In recent literature, methods based on non-convex outer approximation are proposed for tackling one-dimensional equality constraints that are Lipschitz with respect to the maximum norm. To the best of our knowledge, however, there does not exist a non-convex outer approximation method for a general problem class. We introduce a meta-level solution framework to solve such problems and tackle the underlying theoretical foundations. Considering the feasible domain without the constraint function as manageable, our method relaxes the multidimensional constraint and iteratively refines the feasible region by means of norm-induced cuts, relying on an oracle for the resulting subproblems. We show the method’s correctness and investigate the problem complexity. In order to account for discussions about functionality, limits, and extensions, we present computational examples including illustrations.

We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and may result in simpler classes of optimization problems because not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Grant TRR 154].

We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions. The latter occur, for instance, in the context of the curtailment of renewables or the switching of power-generation units and transmission lines. Our approach delivers globally optimal solutions and is provably convergent. We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs (MINLPs). The solution method starts from a known framework that uses piecewise linear relaxations. These relaxations are modeled as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled. In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements. In particular, these are mixed-integer second order cone programs as well as primal and dual cutting planes. For example, objective and no-good cuts help to compute good feasible solutions in which outer approximation constraints tighten the relaxations. We present extensive numerical results for various AC OPF problems in which discrete decisions play a major role. Even for hard instances with a large proportion of discrete decisions, the method is able to generate high-quality solutions efficiently. Furthermore, we compare our approach with state-of-the-art MINLP solvers. Our method outperforms all other algorithms. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research has been funded by the Federal Ministry of Education and Research of Germany [Grant 05M18WEB]. This research has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government. The authors thank the Deutsche Forschungsgemeinschaft for support within projects A05, B06, B07, and B10 of the Sonderforschungsbereich/Transregio 154 “Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks.” This work has been supported by the Federal Ministry for Economic Affairs and Energy, Germany [Grant 03El1036A]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.1270 .

Abstract Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial branch-and-bound become more effective the tighter the used relaxations are. Relaxations are commonly established by convex underestimators, where each constraint function is considered separately. Instead, a considerably tighter relaxation can be found via so-called simultaneous convexification, where convex underestimators are derived for more than one constraint function at a time. In this work, we present a global solution approach for solving mixed-integer nonlinear problems that uses simultaneous convexification. We introduce a separation method that relies on determining the convex envelope of linear combinations of the constraint functions and on solving a nonsmooth convex problem. In particular, we apply the method to quadratic absolute value functions and derive their convex envelopes. The practicality of the proposed solution approach is demonstrated on several test instances from gas network optimization, where the method outperforms standard approaches that use separate convex relaxations.

We propose a decomposition based method for solving mixed‐integer nonlinear optimization problems with “black‐box” nonlinearities, where the latter, for example, may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed‐integer linear master problem and a separation problem for iteratively refining the mixed‐integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with a global optimal solution of the mixed‐integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady‐state gas transport. For the latter we also present promising numerical results of our method applied to real‐world instances that particularly show the effectiveness of our method for problems defined on networks.

In this paper, we study the problem of technical transient gas network optimization, which can be considered a minimum cost flow problem with a nonlinear objective function and additional nonlinear constraints on the network arcs. Applying an implicit box scheme to the isothermal Euler equation, we derive a mixed-integer nonlinear program. This is solved by means of a combination of (i) a novel mixed-integer linear programming approach based on piecewise linearization and (ii) a classical sequential quadratic program applied for given combinatorial constraints. Numerical experiments show that better approximations to the optimal control problem can be obtained by using solutions of the sequential quadratic programming algorithm to improve the mixed-integer linear program. Moreover, iteratively applying these two techniques improves the results even further.

Given a graph $G=(V,E)$ with node weights $\varphi_v \in \mathbb{N}\cup\{0\}$, $v\in V$, and some number $F\in \mathbb{N}\cup\{0\}$, the convex hull of the incidence vectors of all cuts $\delta(S)$, $S\subseteq V$, with $\varphi(S)\le F$ and $\varphi(V\setminus S)\le F$ is called the bisection cut polytope. We study the facial structure of this polytope which shows up in many graph partitioning problems with applications in VLSI design or frequency assignment. We give necessary and in some cases sufficient conditions for the knapsack tree inequalities introduced in [C. E. Ferreira et al., Math. Programming, 74 (1996), pp. 247–267] to be facet-defining. We extend these inequalities to a richer class by exploiting the fact that each cut intersects each cycle in an even number of edges. Finally, we present a new class of inequalities that are based on nonconnected substructures yielding nonlinear right-hand sides. We show that the supporting hyperplanes of the convex envelope of this nonlinear function correspond to the faces of the so-called cluster weight polytope, for which we give a complete description under certain conditions.

We study the parallelization of the steepest-edge version of the dual simplex algorithm. Three different parallel implementations are examined, each of which is derived from the CPLEX dual simplex implementation. One alternative uses PVM, one general-purpose System V shared-memory constructs, and one the PowerC extension of C on a Silicon Graphics multi-processor. These versions were tested on different parallel platforms, including heterogeneous workstation clusters, Silicon Graphics multi-processors, and an IBM SP2. We report on our computational experience.

In this paper, we present the implementation of a branch-and-cut algorithm for solving Steiner tree problems in graphs. Our algorithm is based on an integer programming formulation for directed graphs and comprises preprocessing, separation algorithms, and primal heuristics. We are able to solve nearly all problem instances discussed in the literature to optimality, including one problem that—to our knowledge—has not yet been solved. We also report on our computational experiences with some very large Steiner tree problems arising from the design of electronic circuits. All test problems are gathered in a newly introduced library called SteinLib that is accessible via the World Wide Web. © 1998 John Wiley & Sons, Inc. Networks 32: 207–232, 1998

In this paper, we investigate separation problems for classes of inequalities valid for the polytope associated with the Steiner tree packing problem, a problem that arises, e.g., in very large-scale integration (VLSI) routing. The separation problem for Steiner partition inequalities is $\mathcal{N P}$-hard in general. We show that it can be solved in polynomial time for those instances that come up in switchbox routing. Our algorithm uses dynamic programming techniques. These techniques are also applied to the much more complicated separation problem for alternating cycle inequalities. In this case, we can compute in polynomial time, given some point y, a lower bound for the gap $\alpha - a^T y$ over all alternating cycle inequalities $a^T x \geq \alpha $. This gives rise to a very effective separation heuristic. A by-product of our algorithm is the solution of a combinatorial optimization problem that is interesting in its own right: find a shortest path in a graph where the “length” of a path is its usual length minus the length of its longest edge.