11 research outputs found
Global Optimisation for Energy System
The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complexity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is complicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model discrete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The formulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity
Strengthening SONC Relaxations with Constraints Derived from Variable Bounds
Certificates of polynomial nonnegativity can be used to obtain tight dual
bounds for polynomial optimization problems. We consider Sums of Nonnegative
Circuit (SONC) polynomials certificates, which are well suited for sparse
problems since the computational cost depends only on the number of terms in
the polynomials and does not depend on the degrees of the polynomials. This
work is a first step to integrating SONC-based relaxations of polynomial
problems into a branch-and-bound algorithm. To this end, the SONC relaxation
for constrained optimization problems is extended in order to better utilize
variable bounds, since this property is key for the success of a relaxation in
the context of branch-and-bound. Computational experiments show that the
proposed extension is crucial for making the SONC relaxations applicable to
most constrained polynomial optimization problems and for integrating the two
approaches.Comment: 4 pages, 0 figures, published in proceedings of the Hungarian Global
Optimization Workshop HUGO 202
Efficient Separation of RLT Cuts for Implicit and Explicit Bilinear Products
The reformulation-linearization technique (RLT) is a prominent approach to
constructing tight linear relaxations of non-convex continuous and
mixed-integer optimization problems. The goal of this paper is to extend the
applicability and improve the performance of RLT for bilinear product
relations. First, a method for detecting bilinear product relations implicitly
contained in mixed-integer linear programs is developed based on analyzing
linear constraints with binary variables, thus enabling the application of
bilinear RLT to a new class of problems. Our second contribution addresses the
high computational cost of RLT cut separation, which presents one of the major
difficulties in applying RLT efficiently in practice. We propose a new RLT
cutting plane separation algorithm which identifies combinations of linear
constraints and bound factors that are expected to yield an inequality that is
violated by the current relaxation solution. A detailed computational study
based on implementations in two solvers evaluates the performance impact of the
proposed methods.Comment: 16 pages, 0 figures, submitted to the 24th Conference on Integer
Programming and Combinatorial Optimizatio
Convex Relaxations for Quadratic On/Off Constraints and Applications to Optimal Transmission Switching
Incremental design of water symbiosis networks with prior knowledge: The case of an industrial park in Kenya
The SCIP Optimization Suite 7.0
The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. This paper discusses enhancements and extensions contained in version 7.0 of the SCIP Optimization Suite. The new version features the parallel presolving library PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear optimization problems and can be used stand-alone or integrated into SCIP via a presolver plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides improvements in the Benders’ decomposition solver of SCIP, user-defined decomposition structures can be read, which are used by the automated Benders’ decomposition solver and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation that is used to predict the completion of the overall solving process and potentially trigger restarts. Moreover, substantial performance improvements of the MIP core were achieved by new developments in presolving, primal heuristics, branching rules, conflict analysis, and symmetry handling. Last, not least, the report presents updates to other components and extensions of the SCIP Optimization Suite, in particular, the LP solver SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP
The SCIP Optimization Suite 7.0
The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. This paper discusses enhancements and extensions contained in version 7.0 of the SCIP Optimization Suite. The new version features the parallel presolving library PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear optimization problems and can be used stand-alone or integrated into SCIP via a presolver plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides improvements in the Benders’ decomposition solver of SCIP, user-defined decomposition structures can be read, which are used by the automated Benders’ decomposition solver and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation that is used to predict the completion of the overall solving process and potentially trigger restarts. Moreover, substantial performance improvements of the MIP core were achieved by new developments in presolving, primal heuristics, branching rules, conflict analysis, and symmetry handling. Last, not least, the report presents updates to other components and extensions of the SCIP Optimization Suite, in particular, the LP solver SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP
Enabling Research Through The SCIP Optimization Suite 8.0
The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. The focus of this article is on the role of the SCIP Optimization Suite in supporting research. SCIP's main design principles are discussed, followed by a presentation of the latest performance improvements and developments in version 8.0, which serve both as examples of SCIP's application as a research tool and as a platform for further developments. Furthermore, this article gives an overview of interfaces to other programming and modeling languages, new features that expand the possibilities for user interaction with the framework, and the latest developments in several extensions built upon SCIP.</p