Ivo Nowak

• Prof. Dr.
• Professor at HAW Hamburg

We present a framework for generating reformuations of complex optimization and control problems that can be solved easier than the original problem, called generate-and-solve. The methods are implemented in the open-source frameworks Decogo and Decolearn. Numerical results for nonconvex MINLPs and reinforcement learning are presented.

The use of e-learning systems has a long tradition, where students can study online helped by a system. In this context, the use of recommender systems is relatively new. In our research project, we investigated various ways to create a recommender system. They all aim at facilitating the learning and understanding of a student. We present a common...

We present a novel relaxation for general nonconvex sparse MINLP problems, called overlapping convex hull relaxation (CHR). It is defined by replacing all nonlinear constraint sets by their convex hulls. If the convex hulls are disjunctive, e.g. if the MINLP is block-separable, the CHR is equivalent to the convex hull relaxation obtained by (standa...

In recent years, an interest appeared in integrating various optimization algorithms in machine learning. We study the potential of ensemble learning in classification tasks and how to efficiently decompose the underlying optimization problem. Ensemble learning has become popular for machine learning applications and it is particularly interesting...

We present new decomposition methods for globally solving complex optimization, control and machine learning problems in engineering. The methods are based on a generate-and-solve approach and are implemented in the open-source frameworks Decogo and Decolearn. Numerical results and possible extensions for complex planning, design and control applic...

We present new decomposition methods for globally solving complex optimization and machine learning problems based on a generate-refine-and-solve (GRS) approach using inner and outer approximations. The methods are implemented in the open-source frameworks Decogo and Decolearn. Numerical results for complex nonconvex MINLPs are presented. Furthermo...

Currently minor improvements in image classification tasks are achieved by significantly increasing model complexity. This trend has been ongoing for the last years and high performance models now usually have millions of parameters. Inspired by Ensemble methods, we investigate the potential of Ensemble Learning (EL) to iteratively extend an ensemb...

Several aspects we elaborated in the development of algorithms for MINLP. Mainly the resource constraint vision and the concept of Column Generation are highlighted.

We present inner approximations methods for nonconvex MINLP and Ensemble Learning

Optimization and learning problems are becoming increasingly complex and often cannot be solved with traditional deterministic methods , like Branch-and-Bound (BB). We present inner approximation methods for globally solving complex optimization and learning problems based on generating approximation models (master problems) by solving easier sub-p...

Energy system optimization models are typically large models which combine sub-models which range from linear to very nonlinear. Column generation (CG) is a classical tool to generate feasible solutions of sub-models, defining columns of global master problems, which are used to steer the search for a global solution. In this paper, we present a n...

Most industrial optimization problems are sparse and can be formulated as block-separable mixed-integer nonlinear programming (MINLP) problems, defined by linking low-dimensional sub-problems by (linear) coupling constraints. This paper investigates the potential of using decomposition and a novel multiobjective-based column and cut generation appr...

In this talk we present Decogo, a generic software framework for solving sparse nonconvex MINLPs, based on decomposition based successive approximation. Similar as Column Generation (CG) algorithms for solving huge crew scheduling problems, Decogo computes a solution candidate of a MINLP by first computing a solution of a convex hull relaxation (CR...

Many industrial optimization problems are sparse and can be formulated as block-separable mixed-integer nonlinear programming (MINLP) problems, where low-dimensional sub-problems are linked by a (linear) knapsack-like coupling constraint. This paper investigates exploiting this structure using decomposition and a resource constraint formulation of...

This paper presents a new two-phase method for solving convex mixed-integer nonlinear programming (MINLP) problems, called Decomposition-based Outer Approximation Algorithm (DECOA). In the first phase, a sequence of linear integer relaxed sub-problems (LP phase) is solved in order to rapidly generate a good linear relaxation of the original MINLP p...

In this paper, we present a new multi-tree approach for solving large scale Global Optimization Problems (GOP), called DECOA (Decomposition-based Outer Approximation). DECOA is based on decomposing a GOP into sub-problems, which are coupled by linear constraints. It computes a solution by alternately solving sub- and master-problems using Branch-an...

Most industrial optimization problems are sparse and can be formulated as block-separable mixed-integer nonlinear programming (MINLP) problems, defined by linking low-dimensional sub-problems by (linear) coupling constraints. Decomposition methods solve a block-separable MINLP by alternately solving master problems and sub-problems. In practice, de...

We present new decomposition-based outer and inner approximazion algorithms for solving block-separable MINLPs

Traditional deterministic global optimization methods are often based on a Branch-and-Bound (BB) search tree, which may grow rapidly, preventing the method to find a good solution. Motivated by decomposition-based inner approximation (column generation) methods for solving transport scheduling problems with over 100 million variables, we present tw...

Traditional deterministic global optimization methods are often based on a Branch-and-Bound (BB) search tree, which may grow rapidly, preventing the method to find a good solution. Motivated by decomposition-based inner approximation (column generation) methods for solving transport scheduling problems with over 100 million variables, we present a...

Traditional deterministic global optimization methods are often based on a Branch-and-Bound (BB) search tree, which may grow rapidly, preventing the method to find a good solution. Motivated by decomposition-based inner approximation (column generation) methods for solving transport scheduling problems with over 100 million variables, we present tw...

Traditional deterministic global optimization methods are often based on a (branch-and-bound) search tree, which may grow rapidly, preventing the method to find a good solution. Motivated by decomposition-based inner approximation (column generation) methods for solving transport scheduling problems with over 100 million variables, we present a new...

The Topology Optimization (TO) problem is a basic engineering problem of distributing a limited amount of material in a design space. The finite element formulatuation of TO is a large scale nonconvex Mixed Integer Quadratically Constrained Quadratic Program (MIQQP) with millions of variables. Traditional methods for MIQQP are branch-and-bound meth...

Most industrial optimization problems are sparse, and can be reformulated by smaller sub-problems, which are linked by coupling constraints. Typical examples are planning, control and design problems. In practice, parallel decomposition methods are sometimes the only possibility to compute high-quality solutions of large-scale optimization problems...

The paper examines the applicability of mathematical programming methods to the simultaneous optimization of the structure and the operational parameters of a combined-cycle-based cogeneration plant. Thus, the optimization problem is formulated as a highly non-convex mixed-integer nonlinear problem (MINLP) and solved by the MINLP solver LaGO. The a...

The paper examines the applicability of mathematical programming methods to the simultaneous optimization of the structure and the operational parameters of a combined-cycle-based cogeneration plant. The optimization problem is formulated as a nonconvex mixed-integer nonlinear problem (MINLP) and solved by the MINLP solver LaGO. The algorithm gener...

We propose an efficient column generation method to minimize the prob-ability of delay propagations along aircraft rotations. In this way, delay resistant schedules can be constructed. Computational results for large-scale real-world problems demonstrate substantial punctuality improve-ments. The method can be generalized to crew and integrated sch...

We present a Branch and Cut algorithm of the software package LaGO to solve nonconvex mixed-integer nonlinear programs (MINLPs). A linear outer approximation is constructed from a convex relaxation of the problem. Since we do not require an algebraic representation of the problem, reformulation techniques for the construction of the convex relaxati...

We propose a new scenario tree reduction algorithm for multistage stochastic programs, which integrates the reduction of a scenario tree into the solution process of the stochastic program. This allows to construct a scenario tree that is highly adapted on the optimization problem. The algorithm starts with a rough approximation of the original tre...

We present a Branch and Cut algorithm of the software package LaGO to solve nonconvex mixed-integer nonlinear programs. A linear outer approximation is constructed from a convex relaxation of the problem. Since we do not require an algebraic representation of the problem, reformulation techniques for the construction of the convex relaxation cannot...

Mathematicians and engineers have developed in a joint research project a solution approach for performing simultaneous structural and design variable optimization in the design of cost-effective complex energy conversion systems. The paper presents a methodology and an application to the design of a combined-cycle power plant that provides fixed a...

This paper presents a smoothing heuristic for an NP-hard combinatorial problem. Starting with a convex Lagrangian relaxation, a pathfollowing method is applied to obtain good solutions while gradually transforming the relaxed problem into the original problem formulated with an exact penalty function. Starting points are drawn using different sampl...

A central problem of branch-and-bound methods for global optimization is that lower bounds are often not exact even if the diameter of the subdivided regions shrinks to zero. This can lead to a large number of subdivisions preventing the method from terminating in reasonable time. For the all-quadratic optimization problem with convex constraints w...

In this paper we propose a global optimality criterion for globally minimizing a quadratic form over the standard simplex, which in addition provides a sharp lower bound for the optimal value. The approach is based on the solution of a semidefinite program (SDP) and a convex quadratic program (QP). Since there exist fast (polynomial time) algorithm...

Die Habilitationsschrift beschäftigt sich mit Theorie, Algorithmen und Software zur Lösung von nichtkonvexen, gemischt-ganzzahligen, nichtlinearen Optimierungsproblemen (MINLP). Sie besteht aus 14 Kapiteln, die in zwei Teile gegliedert sind. Im ersten Teil werden grundlegende Optimierungswerkzeuge beschrieben und im zweiten Teil werden Lösungsalgor...

The purpose of this paper is threefold. First we propose splitting schemes for re- formulating non-separable problems as block-separable problems. Second we show that the Lagrangian dual of a block-separable mixed-integer all-quadratic program (MIQQP) can be formulated as an eigenvalue optimization problem keeping the block-separable structure. Fi-...

International Series of Numerical Mathematics Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming 10.1007/3-7643-7374-1_14 IvoNowak 14.LaGO — An Object-Oriented Library for Solving MINLPs

This book presents a comprehensive description of theory, algorithms and software for solving nonconvex mixed integer nonlinear programs (MINLP). The main focus is on deterministic global optimization methods, which play a very important role in integer linear programming, and are used only recently in MINLP. The presented material consists of two...

In this paper we propose a global optimality criterion for globally minimizing a quadratic form over the standard simplex, which in addition provides a sharp lower bound for the optimal value. The approach is based on the solution of a semidefinite program (SDP) and a convex quadratic program (QP). Since there exist fast (polynomial time) algorithm...

The paper describes a software package called LaGO for solving nonconvex mixed integer nonlinear programs (MINLPs). The main component of LaGO is a convex relaxation which is used for generating solution candidates and computing lower bounds of the optimal value. The relaxation is generated by reformulating the given MINLP as a block-separable prob...

The purpose of this paper is threefold. First we show that the Lagrangian dual of a block-separable mixed-integer all-quadratic program (MIQQP) can be formulated as an eigenvalue optimization problem keeping the block-separable structure. Second we propose splitting schemes for reformulating non-separable problems as block-separable problems. Final...

This paper presents smoothing heuristics for an NP-hard combinatorial problem based on Lagrangian relaxation. We formulate the Lagrangian dual for this nonconvex quadratic problem and propose eigenvalue nonsmooth unconstrained optimization to solve the dual problem with bundle or subgradient methods. Derived heuristics are considered to obtain good...

A central problem of branch-and-bound methods for global optimization is that often a lower bound do not match with the optimal value of the corresponding subproblem even if the diameter of the partition set shrinks to zero. This can lead to a large number of subdivisions preventing the method from terminating in reasonable time. For the all-quadra...

A central problem of branch-and-bound methods for global optimization is that lower bounds are often not exact even if the diameter of the subdivided regions shrinks to zero. This can lead to a large number of subdivisions preventing the method from terminating in reasonable time. For the all-quadratic optimization problem with convex constraints w...

The paper describes a method for computing a lower bound of the global minimum of an indefinite quadratic form over a simplex. The bound is derived by computing an underestimator of the convex envelope by solving a semidefinite program (SDP). This results in a convex quadratic program (QP). It is shown that the optimal value of the QP is a lower bo...

In this paper we compare two methods for estimating a global minimizer of an indefinite quadratic form over a simplex. The first method is based on the enumeration of local minimizers of a so-called control polytope. The second method is based on an approximation of the convex envelope using semidefinite programming. In order to test the algorithms...

In this paper we compare two methods for estimating a global minimizer of an indefinite quadratic form over a simplex. The first method is based on the enumeration of local minimizers of a so-called control polytope. The second method is based on an approximation of the convex envelope using semidefinite programming. In order to test the algorithms...

We present a new method, based on generalizations of Shiffman's variational principle, [Nowak 1993; 1994], for the construction of minimal surfaces on Schwarzian chains in curved space forms. The main emphasis of our approach is on the computation of all minimal surfaces of genus zero (disks with holes) that span a given boundary configuration - ev...

We present an algorithm for globally minimizing a multivariate polynomial on a given polyhedron. Our approach is based on the Bernstein-Bezier representation of the polynomial on a simplex. We use global properties of the Bezier points to derive the lower bounds of the global minimum. We discuss the finiteness of the method and present some numeric...

Similar to the investigations of unstable polygonal minimal surfaces by Courant [1] we introduce here a variational principle for the free boundary problem with prescribed topological type which produces minimal surfaces in Riemannian manifolds with constant curvature. For special boundary configurations the surfaces have no branch points. The appr...

Mikrofiche-Ausg.: 2 Mikrofiches : 24x. Berlin, Techn. Universiẗat, Diss., 1994.