Andreas Lundell

Åbo Akademi University · Department of Information Technologies

In this paper, an open-source solver for mixed-integer nonlinear programming (MINLP) problems is presented. The Supporting Hyperplane Optimization Toolkit (SHOT) combines a dual strategy based on polyhedral outer approximations (POA) with primal heuristics. The POA is achieved by expressing the nonlinear feasible set of the MINLP problem with linea...

Different versions of polyhedral outer approximation are used by many algorithms for mixed-integer nonlinear programming (MINLP). While it has been demonstrated that such methods work well for convex MINLP, extending them to solve nonconvex problems has traditionally been challenging. The Supporting Hyperplane Optimization Toolkit (SHOT) is a solve...

The strive for autonomous operation of machines, vehicles and ships requires a leap in the level of self-diagnostics and situation awareness. This self-diagnostic does not directly add value in form of increased performance but is however necessary for safe operations. Hence, there is a need for implementing self-diagnostics systems using non-expen...

In this paper, a recently released open-source solver for convex mixed-integer nonlinear programming (MINLP) is presented. The Supporting Hyperplane Optimization Toolkit (SHOT) solver combines a dual strategy based on polyhedral outer approximations (POA) with several primal heuristics. The outer approximation is achieved by expressing the nonlinea...

Different versions of polyhedral outer approximation is used by many algorithms for mixed-integer nonlinear programming (MINLP). While it has been demonstrated that such methods work well for convex MINLP, extending them to solve also nonconvex problems has been challenging. One solver based on outer linearization of the nonlinear feasible set of M...

The Supporting Hyperplane Optimization Toolkit (SHOT) solver was originally developed for solving convex MINLP problems, for which it has proven to be very efficient. In this paper, we describe some techniques and strategies implemented in SHOT for improving its performance on nonconvex problems. These include utilizing an objective cut to force an...

In this paper, we present a review of deterministic software for solving convex MINLP problems as well as a comprehensive comparison of a large selection of commonly available solvers. As a test set, we have used all MINLP instances classified as convex in the problem library MINLPLib, resulting in a test set of 335 convex MINLP instances. A summar...

In this paper, it is explained how algorithms for convex mixed-integer nonlinear programming (MINLP) based on poly-hedral outer approximation (POA) can be integrated with mixed-integer programming (MIP) solvers through callbacks and lazy constraints. Through this integration, a new approach utilizing a single branching tree is obtained which reduce...

Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optima...

Here we present a center-cut algorithm for convex mixed-integer nonlinear programming (MINLP) that can either be used as a primal heuristic or as a deterministic solution technique. Like several other algorithms for convex MINLP, the center-cut algorithm constructs a linear approximation of the original problem. The main idea of the algorithm is to...

In this paper, a framework for reformulating nonconvex mixed-integer nonlinear programming (MINLP) problems containing twice-differentiable (C²) functions to convex relaxed form is discussed. To provide flexibility and for utilizing more effective transformation strategies, the twice-differentiable functions can be partitioned into convex, signomia...

In this paper a new open source solver for convex mixed-integer nonlinear programming (MINLP) implemented in Wolfram Mathematica is described. The Supporting Hyperplane Optimization Toolkit (SHOT) solver implements two methods for MINLP based on polyhedral outer approximations, namely the Extended Supporting Hyperplane (ESH) and Extended Cutting Pl...

In this paper, we present a new algorithm for solving convex mixed-integer nonlinear programming problems. Similarly to other linearization-based methods, the algorithm generates a polyhedral approximation of the feasible region. The main idea behind the algorithm is to use a different approach for obtaining trial solutions. Here trial solutions ar...

Carbonate rocks are commonly utilized in Wet Flue Gas Desulfurization, WFGD, because of their capability to release calcium ions and precipitate as solid gypsum in an acidic environment. Studies on the reactivity of carbonate rocks and dissolution models can be employed for optimizing the WFGD process. The correct evaluation of limestone reactivity...

In this paper it is shown how to find the guaranteed .-optimal solution to the credibilistic portfolio adjustment problem in the formulation presented by Zhang et al. (2010). In its crisp form, the problem is a non-convex signomial programming problem. This type of problem is difficult to solve to global optimality and solving it using a non-global...

Branch-and-bound in combination with convex underestimators constitute the basis for many algorithms in global nonconvex mixed-integer nonlinear programming (MINLP). Another option is to rely on reformulation-based techniques such as the α signomial global optimization (αSGO) algorithm, where power and exponential transformations for signomial or p...

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or qua...

In this paper, we consider balanced hierarchical data designs for both one-sample and two-sample (two-treatment) location problems. The variances of the relevant estimates and the powers of the tests strongly depend on the data structure through the variance components at each hierarchical level. Also, the costs of a design may depend on the number...

The signomial global optimization algorithm is a method for solving nonconvex mixed-integer signomial problems to global optimality. A convex underestimation is produced by replacing nonconvex signomial terms with convex underestimators obtained through single-variable power and exponential transformations in combination with linearization techniqu...

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are u...

This paper presents an improved as well as a completely new version of a mixed integer linear programming (MILP) formulation for solving the quadratic assignment problem (QAP) to global optimum. Both formulations work especially well on instances where at least one of the matrices is sparse. Modification schemes, to decrease the number of unique el...

The alpha-reformulation (alpha R) technique can be used to transform any nonconvex twice-differentiable mixed-integer nonlinear programming problem to a convex relaxed form. By adding a quadratic function to the nonconvex function it is possible to convexify it, and by subtracting a piecewise linearization of the added function a convex underestima...

In this paper we describe a method for obtaining sets of transformations for reformulating a mixed integer nonlinear programming (MINLP) problem containing nonconvex twice- differentiable (C 2) functions to a convex MINLP problem in an extended variable space. The method for obtaining the transformations is based on solving a mixed integer linear p...

Described in this chapter, is a global optimization algorithm for mixedinteger nonlinear programming problems containing signomial functions. The method obtains a convex relaxation of the nonconvex problem through reformulations using single-variable transformations in combination with piecewise linear approximations of the inverse transformations....

The well-known αBB method solves very general smooth nonconvex optimization problems. The algorithm works by replacing nonconvex functions with convex underestimators. The approximations are improved by branching and bounding until global optimality is achieved. Applications are abundant in engineering and science. We present a convex formulation i...

Convex relaxations play an important role in many areas, especially in optimization and particularly in global optimization. In this paper we will consider some special, but fundamental, issues related to convex relaxation techniques in constrained nonconvex optimization. We will especially consider optimization problems including nonconvex inequal...

In this paper, a method for determining an optimized set of transformations for sig-nomial functions in a nonconvex mixed integer nonlinear programming (MINLP) problem is described. Through the proposed mixed integer linear programming (MILP) problem formulation, a set of single-variable transformations is obtained. By varying the parameters in the...

In this paper, an implementation of a global optimization framework for Mixed Integer Nonlinear Programming (MINLP) problems containing signomial functions is described. In the implementation, the global optimal solution to a MINLP problem is found by solving a sequence of convex relaxed subproblems overestimating the original problem. The describe...

Different types of underestimation strategies are used in deterministic global optimization. In this paper, convexification and underestimation techniques applicable to problems containing signomial functions are studied. Especially, power transformation and exponential transformation (ET) will be considered in greater detail and some new theoretic...

In this paper some transformation techniques, based on power transformations, are discussed. The techniques can be applied to solve optimization problems including signomial functions to global optimality. Signomial terms can always be convexified and underestimated using power transformations on the individual variables in the terms. However, ofte...

Global optimization of mixed integer nonlinear programming (MINLP) problems containing signomial terms is in many cases a difficult task, and many different approaches to solve these problems have been devised. In Westerlund (2005) a method where a relaxed convex relaxation of the original problem is obtained by approximating single-variable transf...

In MINLP (mixed integer nonlinear programming) problems, signomial functions -- especially functions containing bi- and trilinear terms -- are quite common. Problems of this type are generally nonconvex, but it has been shown that it is always possible to write them in a convex form using a two-step transformation technique: In the first step, the...