scholarly journals Embedding Equality Constraints of Optimization Problems into a Quantum Annealer

Algorithms ◽  
2019 ◽  
Vol 12 (4) ◽  
pp. 77 ◽  
Author(s):  
Tomas Vyskocil ◽  
Hristo Djidjev
Keyword(s):  
Optimization Problems ◽  
Energy State ◽  
Minimum Energy ◽  
Combinatorial Design ◽  
Mixed Integer ◽  
Processing Unit ◽  
Alternative Approach ◽  
Quadratic Unconstrained Binary Optimization ◽  
Hard Problems ◽  
Binary Quadratic Optimization

Quantum annealers such as D-Wave machines are designed to propose solutions for quadratic unconstrained binary optimization (QUBO) problems by mapping them onto the quantum processing unit, which tries to find a solution by measuring the parameters of a minimum-energy state of the quantum system. While many NP-hard problems can be easily formulated as binary quadratic optimization problems, such formulations almost always contain one or more constraints, which are not allowed in a QUBO. Embedding such constraints as quadratic penalties is the standard approach for addressing this issue, but it has drawbacks such as the introduction of large coefficients and using too many additional qubits. In this paper, we propose an alternative approach for implementing constraints based on a combinatorial design and solving mixed-integer linear programming (MILP) problems in order to find better embeddings of constraints of the type ∑ x i = k for binary variables x i. Our approach is scalable to any number of variables and uses a linear number of ancillary variables for a fixed k.

scholarly journals An Updated Experimental Evaluation of Graph Bipartization Methods

2021 ◽  
Vol 26 (1) ◽  
pp. 1-24
Author(s):  
Timothy D. Goodrich ◽  
Eric Horton ◽  
Blair D. Sullivan
Keyword(s):  
Optimization Problems ◽  
State Of The Art ◽  
Vertex Cover ◽  
Quantum Annealing ◽  
Heuristic Solution ◽  
Quadratic Unconstrained Binary Optimization ◽  
Odd Cycle ◽  
Iterative Compression ◽  
Hard Problems ◽  
Near Term

We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal (OCT)), motivated by the need for good algorithms for embedding problems into near-term quantum computing hardware. We assemble a preprocessing suite of fast input reduction routines from the OCT and Vertex Cover (VC) literature and compare algorithm implementations using Quadratic Unconstrained Binary Optimization problems from the quantum literature. We also generate a corpus of frustrated cluster loop graphs, which have previously been used to benchmark quantum annealing hardware. The diversity of these graphs leads to harder OCT instances than in existing benchmarks. In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We extend our results with a large corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data. In total, over 8,000 graph instances are evaluated, compared to the previous canonical corpus of 100 graphs. Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.

scholarly journals A Multi-Objective Approach toward Optimal Design of Sustainable Integrated Biodiesel/Diesel Supply Chain Based on First- and Second-Generation Feedstock with Solid Waste Use

Energies ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 2261
Author(s):  
Evgeniy Ganev ◽  
Boyan Ivanov ◽  
Natasha Vaklieva-Bancheva ◽  
Elisaveta Kirilova ◽  
Yunzile Dzhelil
Keyword(s):  
Supply Chain ◽  
Optimal Design ◽  
Solid Waste ◽  
Second Generation ◽  
Agricultural Land ◽  
Optimization Problems ◽  
Biodiesel Production ◽  
Mixed Integer ◽  
Economic Criterion ◽  
Multi Objective

This study proposes a multi-objective approach for the optimal design of a sustainable Integrated Biodiesel/Diesel Supply Chain (IBDSC) based on first- (sunflower and rapeseed) and second-generation (waste cooking oil and animal fat) feedstocks with solid waste use. It includes mixed-integer linear programming (MILP) models of the economic, environmental and social impact of IBDSC, and respective criteria defined in terms of costs. The purpose is to obtain the optimal number, sizes and locations of bio-refineries and solid waste plants; the areas and amounts of feedstocks needed for biodiesel production; and the transportation mode. The approach is applied on a real case study in which the territory of Bulgaria with its 27 districts is considered. Optimization problems are formulated for a 5-year period using either environmental or economic criteria and the remainder are defined as constraints. The obtained results show that in the case of the economic criterion, 14% of the agricultural land should be used for sunflower and 2% for rapeseed cultivation, while for the environmental case, 12% should be used for rapeseed and 3% for sunflower. In this case, the price of biodiesel is 14% higher, and the generated pollutants are 6.6% lower. The optimal transport for both cases is rail.

scholarly journals Nanolaser-based emulators of spin Hamiltonians

Nanophotonics ◽  
2020 ◽  
Vol 9 (13) ◽  
pp. 4193-4198 ◽  
Author(s):  
Midya Parto ◽  
William E. Hayenga ◽  
Alireza Marandi ◽  
Demetrios N. Christodoulides ◽  
Mercedeh Khajavikhan
Keyword(s):  
Large Scale ◽  
Optimization Problems ◽  
Exchange Interactions ◽  
Np Hard ◽  
Spin Models ◽  
Geometric Frustration ◽  
Spin Hamiltonians ◽  
Hard Problems ◽  
On Chip ◽  
Classical Spin Models

AbstractFinding the solution to a large category of optimization problems, known as the NP-hard class, requires an exponentially increasing solution time using conventional computers. Lately, there has been intense efforts to develop alternative computational methods capable of addressing such tasks. In this regard, spin Hamiltonians, which originally arose in describing exchange interactions in magnetic materials, have recently been pursued as a powerful computational tool. Along these lines, it has been shown that solving NP-hard problems can be effectively mapped into finding the ground state of certain types of classical spin models. Here, we show that arrays of metallic nanolasers provide an ultra-compact, on-chip platform capable of implementing spin models, including the classical Ising and XY Hamiltonians. Various regimes of behavior including ferromagnetic, antiferromagnetic, as well as geometric frustration are observed in these structures. Our work paves the way towards nanoscale spin-emulators that enable efficient modeling of large-scale complex networks.

scholarly journals Data Downlink System in the Vast IOT Node Condition Assisted by UAV, Large Intelligent Surface, and Power and Data Beacon

Sensors ◽  
2020 ◽  
Vol 20 (20) ◽  
pp. 5748
Author(s):  
Zhibo Zhang ◽  
Qing Chang ◽  
Na Zhao ◽  
Chen Li ◽  
Tianrun Li
Keyword(s):  
Communication Systems ◽  
Optimization Problems ◽  
Minimum Energy ◽  
Numerical Algorithms ◽  
Flight Trajectory ◽  
Performance Improvements ◽  
Minimum Energy Consumption ◽  
Significant Performance ◽  
Downlink System ◽  
The Internet Of Things

The future development of communication systems will create a great demand for the internet of things (IOT), where the overall control of all IOT nodes will become an important problem. Considering the essential issues of miniaturization and energy conservation, in this study, a new data downlink system is designed in which all IOT nodes harvest energy first and then receive data. To avoid the unsolvable problem of pre-locating all positions of vast IOT nodes, a device called the power and data beacon (PDB) is proposed. This acts as a relay station for energy and data. In addition, we model future scenes in which a communication system is assisted by unmanned aerial vehicles (UAVs), large intelligent surfaces (LISs), and PDBs. In this paper, we propose and solve the problem of determining the optimal flight trajectory to reach the minimum energy consumption or minimum time consumption. Four future feasible scenes are analyzed and then the optimization problems are solved based on numerical algorithms. Simulation results show that there are significant performance improvements in energy/time with the deployment of LISs and reasonable UAV trajectory planning.

ALPHA-HELIX FORMATION IN C-PEPTIDE RNASE-A INVESTIGATED BY PARALLEL TEMPERING SIMULATIONS

2007 ◽  
Vol 18 (01) ◽  
pp. 91-98 ◽  
Author(s):  
GÖKHAN GÖKOĞLU ◽  
TARIK ÇELİK
Keyword(s):  
Energy State ◽  
Minimum Energy ◽  
Alpha Helix ◽  
Salt Bridge ◽  
Rnase A ◽  
Parallel Tempering ◽  
Implicit Solvent ◽  
C Peptide ◽  
Helix Formation ◽  
Α Helix

We have performed parallel tempering simulations of a 13-residue peptide fragment of ribonuclease-A, c-peptide, in implicit solvent with constant dielectric permittivity. This peptide has a strong tendency to form α-helical conformations in solvent as suggested by circular dichroism (CD) and nuclear magnetic resonance (NMR) experiments. Our results demonstrate that 5th and 8–12 residues are in the α-helical region of the Ramachandran map for global minimum energy state in solvent environment. Effects of salt bridge formation on stability of α-helix structure are discussed.

Optimization of Real Power Loss and Voltage Stability Index for Distribution Systems with Distributed Generation

2017 ◽  
Vol 33 ◽  
pp. 100-114 ◽  
Author(s):  
Mostafa Elshahed ◽  
Mahmoud Dawod ◽  
Zeinab H. Osman
Keyword(s):  
Genetic Algorithm ◽  
Distributed Generation ◽  
Distribution Systems ◽  
Voltage Stability ◽  
Reactive Power ◽  
Optimization Problems ◽  
Stability Index ◽  
Mixed Integer ◽  
Voltage Profile ◽  
Voltage Stability Index

Integrating Distributed Generation (DG) units into distribution systems can have an impact on the voltage profile, power flow, power losses, and voltage stability. In this paper, a new methodology for DG location and sizing are developed to minimize system losses and maximize voltage stability index (VSI). A proper allocation of DG has to be determined using the fuzzy ranking method to verify best compromised solutions and achieve maximum benefits. Synchronous machines are utilized and its power factor is optimally determined via genetic optimization to inject reactive power to decrease system losses and improve voltage profile and VSI. The Augmented Lagrangian Genetic Algorithm with nonlinear mixed-integer variables and Non-dominated Sorting Genetic Algorithm have been implemented to solve both single/multi-objective function optimization problems. For proposed methodology effectiveness verification, it is tested on 33-bus and 69-bus radial distribution systems then compared with previous works.

scholarly journals Partially distributed outer approximation

2021 ◽  
Author(s):  
Alexander Murray ◽  
Timm Faulwasser ◽  
Veit Hagenmeyer ◽  
Mario E. Villanueva ◽  
Boris Houska
Keyword(s):  
Large Scale ◽  
Optimization Problems ◽  
Outer Approximation ◽  
Mixed Integer ◽  
Global Optimality ◽  
Integer Optimization ◽  
Global Minimizers ◽  
Outer Approximation Method ◽  
Coupling Constraints ◽  
Outer Approximation Algorithm

AbstractThis paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression.

An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and its Applications to Mechanical Design

1993 ◽  
Author(s):  
B. K. Kannan ◽  
Steven N. Kramer
Keyword(s):  
Lagrange Multiplier ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Optimization Problems ◽  
Mechanical Design ◽  
Continuous Optimization ◽  
Mixed Integer ◽  
Continuous Variables ◽  
Augmented Lagrange Multiplier

Abstract An algorithm for solving nonlinear optimization problems involving discrete, integer, zero-one and continuous variables is presented. The augmented Lagrange multiplier method combined with Powell’s method and Fletcher & Reeves Conjugate Gradient method are used to solve the optimization problem where penalties are imposed on the constraints for integer / discrete violations. The use of zero-one variables as a tool for conceptual design optimization is also described with an example. Several case studies have been presented to illustrate the practical use of this algorithm. The results obtained are compared with those obtained by the Branch and Bound algorithm. Also, a comparison is made between the use of Powell’s method (zeroth order) and the Conjugate Gradient method (first order) in the solution of these mixed variable optimization problems.

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