Large Optimization Problems Research Articles

Optimization of building form to reduce incident solar radiation

In recent years, concerns about global warming have encouraged researchers to incorporate optimization techniques into the design of energy-efficient buildings. Designing a building with energy consumption in mind, design teams should apply the appropriate considerations at the early architectural conceptual design stage. Furthermore, constructing buildings that are energy-efficient results in using less natural resources to cool buildings. Since the impact of solar radiation is a key factor in building energy, reducing insolation (incident solar radiation) is critical for energy-efficient buildings. To alleviate the energy concerns in energy-efficient buildings, this paper proposes an optimization approach to study energy-efficient building form that minimizes insolation while preserving the required total built area. Furthermore, a software system is built with the optimization in the back end and a user interface for experimenting with different design parameters and visualizing the resulting building form. The tool can be used by architectural design teams to design energy-efficient buildings. Due to the large number of design variables in a building and the nonlinear constrained characteristics, the optimization uses the computationally efficient Penalty Successive Linear Programming (PSLP) technique. PSLP is used to solve the large nonlinear optimization problem via a sequence of Linear Programs (LPs) to generate the optimized building form with low computational expense. The optimization and software system are illustrated using real scenarios from several countries while considering different building laws. A complete validation process is undertaken using the CPLEX and MATLAB software. Results show that the optimization tool provides up to 48% less insolation while still meeting different site and building constraints.

ReLU networks as surrogate models in mixed-integer linear programs

We consider the embedding of piecewise-linear deep neural networks (ReLU networks) as surrogate models in mixed-integer linear programming (MILP) problems. A MILP formulation of ReLU networks has recently been applied by many authors to probe for various model properties subject to input bounds. The formulation is obtained by programming each ReLU operator with a binary variable and applying the big-M method. The efficiency of the formulation hinges on the tightness of the bounds defined by the big-M values. When ReLU networks are embedded in a larger optimization problem, the presence of output bounds can be exploited in bound tightening. To this end, we devise and study several bound tightening procedures that consider both input and output bounds. Our numerical results show that bound tightening may reduce solution times considerably, and that small-sized ReLU networks are suitable as surrogate models in mixed-integer linear programs.

Efficient partition of integer optimization problems with one-hot encoding

Quantum annealing is a heuristic algorithm for solving combinatorial optimization problems, and hardware for implementing this algorithm has been developed by D-Wave Systems Inc. The current version of the D-Wave quantum annealer can solve unconstrained binary optimization problems with a limited number of binary variables. However, the cost functions of several practical problems are defined by a large number of integer variables. To solve these problems using the quantum annealer, integer variables are generally binarized with one-hot encoding, and the binarized problem is partitioned into small subproblems. However, the entire search space of the binarized problem is considerably larger than that of the original integer problem and is dominated by infeasible solutions. Therefore, to efficiently solve large optimization problems with one-hot encoding, partitioning methods that extract subproblems with as many feasible solutions as possible are required. In this study, we propose two partitioning methods and demonstrate that they result in improved solutions.

Distributed multi-objective evolutionary optimization using island-based selective operator application

Parallel island-model co-evolutionary algorithms are well-known methods, suitable for dealing with large multi-objective optimization problems. This paper proposes a version of these algorithms where each island modifies a fragment of the chromosome that encodes a possible solution to the problem. The objective of this paper is to demonstrate that automatically setting the size of the overlapping fragments depending on the number of islands obtains better results than using a fixed overlapping size. This method has been compared to other parallel evolutionary techniques considering a different number of islands, chromosome sizes and benchmarks. The analysis of the obtained experimental results, by using different metrics, shows that our approach can provide statistically significant improvements with respect to the base algorithm in high-dimensional, un-decomposable, multi-objective problems. This opens a very promising line to automatically adapt the overlapping sizes in this kind of algorithms.

Pr x Ca 1 − x MnO 3 based stochastic neuron for Boltzmann machine to solve “maximum cut” problem

The neural network enables efficient solutions for Nondeterministic Polynomial-time (NP) hard problems, which are challenging for conventional von Neumann computing. The hardware implementation, i.e., neuromorphic computing, aspires to enhance this efficiency by custom hardware. Particularly, NP hard graphical constraint optimization problems are solved by a network of stochastic binary neurons to form a Boltzmann Machine (BM). The implementation of stochastic neurons in hardware is a major challenge. In this work, we demonstrate that the high to low resistance switching (set) process of a PrxCa1−xMnO3 (PCMO) based RRAM (Resistive Random Access Memory) is probabilistic. Additionally, the voltage-dependent probability distribution approximates a sigmoid function with 1.35%–3.5% error. Such a sigmoid function is required for a BM. Thus, the Analog Approximate Sigmoid (AAS) stochastic neuron is proposed to solve the maximum cut—an NP hard problem. It is compared with Digital Precision-controlled Sigmoid (DPS) implementation using (a) pure CMOS design and (b) hybrid (RRAM integrated with CMOS). The AAS design solves the problem with 98% accuracy, which is comparable with the DPS design but with 10× area and 4× energy advantage. Thus, ASIC neuro-processors based on novel analog neuromorphic devices based BM are promising for efficiently solving large scale NP hard optimization problems.

Newton Method with AOR Iteration for Finding Large Scale Unconstrained Minimizer with Tridiagonal Hessian Matrices

Finding the large scale unconstrained minimizer using Newton method has required the calculation of large and complicated linear systems results from solving the Newton direction. Therefore, in this paper, we propose a method for solving large scale unconstrained optimization problems with tridiagonal Hessian matrices to reduce the complexity of calculating Newton direction. Our proposed method was a combination of Newton method and Accelerated Over Relaxation (AOR) iterative method. To evaluate the performance of the proposed method, combination of Newton method with Gauss-Seidel iteration and Newton method with Successive Over Relaxation (SOR) iteration were used as reference method. Finally, the numerical experiment illustrated that the proposed method produce results that are more efficient compared to the reference methods with less execution time and minimum number of iterations.

Model complex control CMA-ES

Abstract Covariance Matrix Adaptation Evolution Strategy (CMA-ES) has shown great performance on nonseparable optimization problems largely due to its rotation-invariant feature. However, as the computational cost of the self-adaption operation is sensitive to the scale of problems, the performance of CMA-ES heavily suffers from the well-known curse of dimensionality, which makes it impractical to many Large Scale Global Optimization (LSGO) problems. In this paper, a correlation coefficient based grouping (CCG) strategy is proposed to detect the correlations between variables in a simple yet efficient way. Then coupled with a model complexity control (MCC) framework, a new variant of CMA-ES, named MCC-CCG-CMAES, is presented for LSGO problems, which suffers less from curse of dimensionality and significantly reduces the computational cost compared with the standard CMA-ES. To the best of our knowledge, this work is the first attempt at enhancing CMA-ES with the MCC framework rather than the cooperative coevolution (CC) framework. Experimental results on the CEC′2010 large-scale global optimization (LSGO) benchmark functions show that the performance of MCC-CCG-CMAES outperforms the state-of-the-art counterparts.

Application of Genetic Algorithm in Common Optimization Problems

Use of non-deterministic algorithms for solving multi-variable optimization problems is widely used nowadays. Genetic Algorithm belongs to a group of stochastic biomimicry algorithms, it allows us to achieve optimal or near-optimal results in large optimization problems in exceptionally short time (compared to standard optimization methods). Major advantage of Genetic Algorithm is the ability to fuse genes, to mutate and do selection based on fitness parameter. These methods protect us from being trapped in local optima (Most of deterministic algorithms are prone to getting stuck on local optima). In this paper we experimentally show the upper hand of Genetic Algorithms compared to other traditional optimization methods by solving complex optimization problem.

Solving utility-maximization selection problems with Multinomial Logit demand: Is the First-Choice model a good approximation?

We investigate First-Choice (FC) assignment models, a simple type of choice model where customers are allocated to their highest utility option, as a heuristic or starting point for the Multinomial Logit (MNL) model in the context of selection problems with a utility maximization objective. This type of problem occurs in a variety of applications, from location problems to assortment planning or transportation planning. FC assignment models are less refined but computationally more tractable than the more commonly used MNL. MNL suffers from tractability issues due to its nonlinear structure when used within a large size optimization problem with binary decision variables. We design the first comparison of the two modeling frameworks in a context of customer utility maximization for selection problems with binary variables. We provide a probabilistic analysis of the expected customer choice probabilities, document the computational challenges faced by the MNL model in our setting and show in numerical experiments that the FC model exhibits excellent performance as an approximation of the MNL model with an average gap for instance of at most 2.2% for uniformly distributed utilities and of at most 1.4% for normally distributed utilities (and below 1% in a majority of test cases). The key contribution of this paper is to build the case for the FC model as a tractable, high-quality approximation of the MNL model for binary selection problems with utility maximization.

Multi-objective optimization of water injection in spark-ignition engines using the stochastic reactor model with tabulated chemistry

Water injection is investigated for turbocharged spark-ignition engines to reduce knock probability and enable higher engine efficiency. The novel approach of this work is the development of a simulation-based optimization process combining the advantages of detailed chemistry, the stochastic reactor model and genetic optimization to assess water injection. The fast running quasi-dimensional stochastic reactor model with tabulated chemistry accounts for water effects on laminar flame speed and combustion chemistry. The stochastic reactor model is coupled with the Non-dominated Sorting Genetic Algorithm to find an optimum set of operating conditions for high engine efficiency. Subsequently, the feasibility of the simulation-based optimization process is tested for a three-dimensional computational fluid dynamic numerical test case. The newly proposed optimization method predicts a trade-off between fuel efficiency and low knock probability, which highlights the present target conflict for spark-ignition engine development. Overall, the optimization shows that water injection is beneficial to decrease fuel consumption and knock probability at the same time. The application of the fast running quasi-dimensional stochastic reactor model allows to run large optimization problems with low computational costs. The incorporation with the Non-dominated Sorting Genetic Algorithm shows a well-performing multi-objective optimization and an optimized set of engine operating parameters with water injection and high compression ratio is found.

Integrated condition-based track maintenance planning and crew scheduling of railway networks

We develop a multi-level decision making approach for optimal condition-based maintenance planning of a railway network divided into a large number of sections with independent stochastic deterioration dynamics. At higher level, a chance-constrained Model Predictive Control (MPC) controller determines the long-term section-wise maintenance plan, minimizing condition deterioration and maintenance costs for a finite planning horizon, while ensuring that the deterioration level of each section stays below the maintenance threshold with a given probabilistic guarantee in the presence of parameter uncertainty. The resulting large MPC optimization problem containing both continuous and discrete decision variables is solved using Dantzig-Wolfe decomposition to improve the scalability of the proposed approach. At a lower level, the optimal short-term scheduling of the maintenance interventions suggested by the high-level controller and the optimal routing of the corresponding maintenance crew is formulated as a capacitated arc routing problem, which is solved exactly by transforming it into a node routing problem. The proposed approach is illustrated by a numerical case study on the optimal treatment of squats of a regional Dutch railway network. Simulation results show that the proposed approach is robust, non-conservative, and scalable.

A Nonlinear Conjugate Gradient Algorithm under Strong Wolfe-Powell Line Search for Large Scale Unconstrained Optimization Problems

The conventional conjugate gradient method solves linear and quadratic optimization problems but most real life problems consist of nonquadratic functions of several variables. In this work a nonlinear conjugate gradient algorithm for solving large scale optimization problems is presented. The new algorithm is a modification of the Fletcher-Reeves conjugate gradient method and it is proved to achieve global convergence under the strong Wolfe-Powell inexact line search technique. Computational experiments show that the new algorithm presented performs better than the Fletcher-Reeves exact line search algorithm in solving high dimensional nonlinear optimization problems.

Hybrid parallel multimethod hyperheuristic for mixed-integer dynamic optimization problems in computational systems biology

This paper describes and assesses a parallel multimethod hyperheuristic for the solution of complex global optimization problems. In a multimethod hyperheuristic, different metaheuristics cooperate to outperform the results obtained by any of them isolated. The results obtained show that the cooperation of individual parallel searches modifies the systemic properties of the hyperheuristic, achieving significant performance improvements versus the sequential and the non-cooperative parallel solutions. Here we present and evaluate a hybrid parallel scheme of the multimethod, using both message-passing (MPI) and shared memory (OpenMP) models. The hybrid parallelization allows to achieve a better trade-off between performance and computational resources, through a compromise between diversity (number of islands) and intensity (number of threads per island). For the performance evaluation, we considered the general problem of reverse engineering nonlinear dynamic models in systems biology, which yields very large mixed-integer dynamic optimization problems. In particular, three very challenging problems from the domain of dynamic modeling of cell signaling were used as case studies. In addition, experiments have been carried out in a local cluster, a large supercomputer and a public cloud, to show the suitability of the proposed solution in different execution platforms.

Testing evolutionary algorithms for optimization of water distribution networks

Water distribution networks (WDNs) are one of the most important elements of urban infrastructure and require large investment for construction. Design of WDNs is classified as a large combinatorial discrete nonlinear optimization problem. The main concerns associated with the optimization of such networks are the nonlinearity of the discharge-head loss relationships for pipes and the discrete nature of pipe sizes. Due to these issues, this problem is widely considered to be a benchmark problem for testing and evaluating the performance of nonlinear and heuristic optimization algorithms. This paper compares different techniques, all based on evolutionary algorithms (EAs), which yield optimal solutions for least-cost design of WDNs. All of these algorithms search for the global optimum starting from populations of solutions, rather than from a single solution, as in Newton-based search methods. They use different operators to improve the performance of many solutions over repeated iterations. Ten EAs, four of them for the first time, are applied to the design of three networks and their performance in terms of the least cost, under different stopping criteria, are evaluated. Statistical information for 20 executions of the ten algorithms is summarized, and Friedman tests are conducted. Results show that, for the two-loop benchmark network, the particle swarm optimization gravitational search and biology and bioinformatics global optimization algorithms efficiently converge to the global optimum, but perform poorly for large networks. In contrast, given a sufficient number of function evaluations, the covariance matrix adaptation evolution strategy and soccer league competition algorithm consistently converge to the global optimum, for large networks.

A game theory-based decentralized control strategy for power demand management of building cluster using thermal mass and energy storage

The development of smart grids requires more active and effective participation of buildings in power balance. However, most of building demand management and demand response control strategies focus on single buildings only. For a group of buildings at cluster-level, which are often involved in an electricity charge account, such control strategies will not be effective. A game theory-based decentralized control strategy is therefore developed to address the demand management of cluster-level buildings. The indoor temperature set-point and the charging/discharging process of active cold storages in central air-conditioning systems are optimized simultaneously. Rather than optimizing the power demand of all buildings on a central optimization system, the proposed strategy optimizes the power demand of all buildings collectively in a decentralized manner. Using this strategy, buildings manage their own power demands locally only using the aggregated power demand of building cluster as the common reference for their demand controls. This distributed computing allows the optimization of large systems or complex optimization problems to be divided into a few simple optimization tasks, providing enhanced applicability and robustness in practical applications. Case studies are conducted and results show that the proposed game theory-based decentralized control strategy can increase the aggregated peak demand reduction and electricity cost saving more than two times compared with that when the demand management of building cluster is conducted in an uncoordinated manner. Meanwhile, the control performance of proposed decentralized strategy is close to that using a perfect demand management control strategy.

A class of accelerated conjugate-gradient-like methods based on a modified secant equation

This paper proposes a new class of accelerated conjugate-gradient-like algorithms for solving large scale unconstrained optimization problems, which combine the idea of accelerated adaptive Perry conjugate gradient algorithms proposed by Andrei (2017) with the modified secant condition and the nonmonotone line search technique. An attractive property of the proposed methods is that the search direction always provides sufficient descent step which is independent of the line search used and the convexity of objective function. Under common assumptions, it is proven that the proposed methods possess global convergence for nonconvex smooth functions, and R-linear convergence for uniformly convex functions. The numerical experiments show the efficiency of the proposed method in practical computations.

Clustering methods for large scale geometrical global optimization

ABSTRACTIn this paper we show that for some problem classes it is possible to design Global Optimization algorithms which mimic existing procedures obtaining the same quality at a fraction of their computational cost. We achieved this applying clustering methods to identify regions of attraction of local minima. If we could identify the shape of regions of attraction, a local search starting from each of them would lead to the global minimum. This idea had been a winning one in the 1980s, and later abandoned when large dimensional global optimization problems were used to test global optimization algorithms. In this paper we show that by using the idea of clustering in a feature space of much smaller dimension than the original one, very significant speed ups can be obtained. We apply this idea to two of the most widely studied families of hard, large scale, global optimization problems: the optimization of the potential energy of atomic clusters, and the problem of packing identical spheres of largest radius in the unit hypercube. We could even improve some existing putative optima, thus proving that the proposed method is not only very efficient but also effective in exploring the feasible space.

A Bayesian Stochastic Kriging Optimization Model Dealing with Heteroscedastic Simulation Noise for Freeway Traffic Management

Since advanced traveler information and traffic management systems have become popular, it is vital to capture the joint impact of various strategies on transportation systems. Developing analytical models to incorporate traffic dynamics and travel behavior is challenging. Based on observation of the heteroscedasticity of the simulation noise in the stochastic simulator, this paper develops the Bayesian stochastic Kriging (BSK) model to adapt for the heteroscedastic noise and metamodel parameter uncertainty in a Bayesian framework, which enhances the existing surrogate-based optimization methods. This paper presents a metamodel for large scale simulation-based freeway traffic management optimization problems. Simulation-based optimization combines the advantages of simulation models and mathematical optimization methods. The parameter estimation of the BSK model is accomplished by the Bayesian inference. The proposed methodology enables the efficient use of large scale high-resolution traffic simulation models for simulation-based optimization, while accounting for travelers’ behavioral responses to information provision. We demonstrate the advantages of BSK compared to other existing metamodels using a numerical example of a synthetic network and a mathematical example. In a work zone scenario on a real-world freeway/arterial corridor of I-270 and MD-355 in the State of Maryland, the BSK model is applied to the freeway traffic management via optimizing the high-occupancy/toll rate and deploying dynamic message signs. Field traffic measurements by loop/microwave detections are used to calibrate travel demand and to supply the simulation parameters. The optimization results are promising in reducing the corridor-wide travel delay and enhancing the vehicle throughput. The online appendix is available at https://doi.org/10.1287/trsc.2018.0819 .

Multi-Level Energy Management for Hybrid Electric Vehicles—Part I

The fuel economy of a hybrid electric vehicle (HEV) is improved, by taking the energy relevant system states into account in the energy management system (EMS). With an increasing number of states and decision variables, energy optimizing algorithms in the EMS can be prohibitive for real-time implementation. In part I of this work, a model-based, multi-level approach is taken to subdivide the original (large) optimization problem into computational efficient sub-problems, based on optimal control techniques using a preview. The resulting EMS solves the problem of power-split between engine and motor/generator, mode and gear switching including switching costs, with battery energy constraints. The superior energy efficiency of the multi-level EMS is simulated on a representative heavy duty drive cycle, where it saves 7.0% fuel, compared to a conventional vehicle, where the baseline EMS for the HEV saves 5.8%. In part II, real-world validation of the EMS is performed.

Optimal operation of dynamic (energy) systems: When are quasi-steady models adequate?

Since design optimization faces the challenge of solving inherently large optimization problems, the complexity of underlying dynamic systems is often reduced by applying quasi-steady state assumptions. It is thereby indispensable to identify the components whose transient behavior is essential to ensure meaningful results. In this study, we discuss a dynamic model for an illustrative hybrid energy system, which extends the quasi-steady models of Voll et al. (2013). Based on optimal operation with fixed design, we underline the importance of the relationship between the dynamics of the model and of the input data for the adequateness of quasi-steady operation. Our results emphasize the need for suitable ramp constraints in quasi-steady models of dynamic systems within operational optimization. Moreover, the existence of a storage unit is no sufficient justification for quasi-steady state assumptions.