A Finite-Time Cutting Plane Algorithm for Distributed Mixed Integer Linear Programming

Many problems of interest for cyber-physical network systems can be formulated as Mixed Integer Linear Programs in which the constraints are distributed among the agents. In this paper we propose a distributed algorithm to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. In the proposed algorithm, at each communication round, agents solve locally a small LP, generate suitable cutting planes, namely intersection cuts and cost-based cuts, and communicate a fixed number of active constraints, i.e., a candidate optimal basis. We prove that, if the cost is integer, the algorithm converges to the lexicographically minimal optimal solution in a finite number of communication rounds. Finally, through numerical computations, we analyze the algorithm convergence as a function of the network size.Comment: 6 pages, 3 figure

Developing an Enhanced Algorithms to Solve Mixed Integer Non-Linear Programming Problems Based on a Feasible Neighborhood Search Strategy

Engineering optimization problems often involve nonlinear objective functions, which can capture complex relationships and dependencies between variables. This study focuses on a unique nonlinear mathematics programming problem characterized by a subset of variables that can only take discrete values and are linearly separable from the continuous variables. The combination of integer variables and non-linearities makes this problem much more complex than traditional nonlinear programming problems with only continuous variables. Furthermore, the presence of integer variables can result in a combinatorial explosion of potential solutions, significantly enlarging the search space and making it challenging to explore effectively. This issue becomes especially challenging for larger problems, leading to long computation times or even infeasibility. To address these challenges, we propose a method that employs the "active constraint" approach in conjunction with the release of nonbasic variables from their boundaries. This technique compels suitable non-integer fundamental variables to migrate to their neighboring integer positions. Additionally, we have researched selection criteria for choosing a nonbasic variable to use in the integerizing technique. Through implementation and testing on various problems, these techniques have proven to be successful

MINLP optimization of structures

The paper discusses the mixed-integer non-linear programming (MINLP) approach to the optimization of structures. The MINLP is an optimization technique which is able to solve non-linear and discrete optimization problems. It calculates continuous variables (loads, dimensions, stresses, deflections, costs) and discrete variables (topology, standard sections, material grades). The MINLP optimization model of a structure should be developed. In the model, an objective function is subjected to structural analysis and dimensioning constraints in order to satisfied ultimate and serviceability limit states according to Eurocodes. Appropriate MINLP algorithms and strategies are used to solve the defined MINLP problem. Two numerical examples are presented at the end of the paper

Stochastic Cutting Planes for Data-Driven Optimization

We introduce a stochastic version of the cutting-plane method for a large class of data-driven Mixed-Integer Nonlinear Optimization (MINLO) problems. We show that under very weak assumptions the stochastic algorithm is able to converge to an $\epsilon$-optimal solution with high probability. Numerical experiments on several problems show that stochastic cutting planes is able to deliver a multiple order-of-magnitude speedup compared to the standard cutting-plane method. We further experimentally explore the lower limits of sampling for stochastic cutting planes and show that for many problems, a sampling size of $O(\sqrt[3]{n})$ appears to be sufficient for high quality solutions

Pup Matching: Model Formulations and Solution Approaches

We model Pup Matching, the logistics problem of matching or pairing semitrailers known as pups to cabs that are able to tow one or two of the pups simultaneously, as an AfP-complete version of the Network Loading Problem (NLP). We examine a branch and bound solution approach tailored to the NLP formulation through the use of three families of cutting planes and four heuristic procedures. Theoretically, we specify facet defining conditions for a cut family that we refer to as odd flow inequalities and show that each heuristic yields a 2-approximation. Computationally, the cheapest of the four heuristic values achieved an average error of 1.3% among solved test problems randomly generated from realistic data. Branch and bound solved to optimality 67% of these problems. Application of the cutting plane families reduced the average relative difference between upper and lower bounds prior to branching from 18.8% to 6.4%

Linear programming on the Stiefel manifold

Linear programming on the Stiefel manifold (LPS) is studied for the first time. It aims at minimizing a linear objective function over the set of all $p$-tuples of orthonormal vectors in ${\mathbb R}^n$ satisfying $k$ additional linear constraints. Despite the classical polynomial-time solvable case $k=0$, general (LPS) is NP-hard. According to the Shapiro-Barvinok-Pataki theorem, (LPS) admits an exact semidefinite programming (SDP) relaxation when $p(p+1)/2\le n-k$, which is tight when $p=1$. Surprisingly, we can greatly strengthen this sufficient exactness condition to $p\le n-k$, which covers the classical case $p\le n$ and $k=0$. Regarding (LPS) as a smooth nonlinear programming problem, we reveal a nice property that under the linear independence constraint qualification, the standard first- and second-order {\it local} necessary optimality conditions are sufficient for {\it global} optimality when $p+1\le n-k$

MINLP OPTIMIZATION OF STEEL FRAMES

ABSTRACT: The paper presents the discrete dimension optimization of unbraced rigid steel plane frames. The optimization of steel frames was carried out by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLP is a combined discrete-continuous optimization technique. It performs the discrete optimization of discrete decisions simultaneously with the continuous optimization of continuous parameters. The task of the optimization is to minimize the mass of the frame structure and to find the optimal discrete sizes of standard steel sections for frame members. The finite element equations are defined as the equality constraints for the second-order elastic structural analysis. The design constraints for the steel members were formulated according to Eurocode 3. The Modified Outer-Approximation/ Equality-Relaxation algorithm and a two-phase MINLP optimization approach were applied for the optimization. The latter starts with the continuous optimization of the frame, while the standard dimensions are temporarily relaxed into continuous parameters. When the optimal continuous solution is found, standard sizes of cross-sections are re-established and the simultaneous continuous and discrete dimension optimization by MINLP is then continued until the optimal solution is found. A numerical example of the optimization of a steel frame is presented at the end of the paper to show the suitability of the proposed approach

Structure Exploitation in Mixed-Integer Optimization with Applications to Energy Systems

Das Ziel dieser Arbeit ist neue numerische Methoden für gemischt-ganzzahlige Optimierungsprobleme zu entwickeln um eine verbesserte Geschwindigkeit und Skalierbarkeit zu erreichen. Dies erfolgt durch Ausnutzung gängiger Problemstrukturen wie separierbarkeit oder Turnpike-eigenschaften. Methoden, die diese Strukturen ausnutzen können, wurden bereits im Bereich der verteilten Optimierung und optimalen Steuerung entwickelt, sie sind jedoch nicht direkt auf gemischt-ganztägige Probleme anwendbar. Um verteilte Rechenressourcen zur Lösung von gemischt-ganzzahligen Problemen nutzen zu können, sind neue Methoden erforderlich. Zu diesem Zweck werden verschiedene Erweiterungen bestehender Methoden sowie neuartige Techniken zur gemischt-ganzzahligen Optimierung vorgestellt. Benchmark-Probleme aus Strom- und Energiesystemen werden verwendet, um zu demonstrieren, dass die vorgestellten Methoden zu schnelleren Laufzeiten führen und die Lösung großer Probleme ermöglichen, die sonst nicht zentral gelöst werden können. Die vorliegende Arbeit enthält die folgenden Beiträge: - Eine Erweiterung des Augmented Lagrangian Alternating Direction Inexact Newton-Algorithmus zur verteilten Optimierung für gemischt-ganzzahlige Probleme. - Ein neuer, teilweise-verteilter Optimierungsalgorithmus für die gemischt-ganzzahlige Optimierung basierend auf äußeren Approximationsverfahren. - Ein neuer Optimierungsalgorithmus für die verteilte gemischt-ganzzahlige Optimierung, der auf branch-and-bound Verfahren basiert. - Eine erste Untersuchung von Turnpike-Eigenschaften bei Optimalsteuerungsproblemen mit gemischten-Ganzzahligen Entscheidungsgrößen und ein spezieller Algorithmus zur Lösung dieser Probleme. - Eine neue Branch-and-Bound Heuristik, die a priori Probleminformationen effizienter nutzt als aktuelle Warmstarttechniken. Schließlich wird gezeigt, dass die Ergebnisse der vorgestellten Optimierungsalgorithmen für verteilte gemischt-ganzzahlige Optimierung stark Partitionierungsabhängig sind. Zu diesem Zweck wird auch eine Untersuchung von Partitionierungsmethoden für die verteilte Optimierung vorgestellt