Mixed-integer Optimization Approach Research Articles

Optimal Generation Capacity Allocation and Droop Control Design for Current Sharing in DC Microgrids

Considering the increasing amounts of renewable energy generation and energy storage required to meet ambitious environmental goals, it is clear that the next generation of power grids will be dominated by converter-connected devices. In addition, the increasing share of loads connected via power electronics and the general transition to non-synchronous grids with distributed generators make dc microgrids an attractive future alternative. However, achieving optimal utilization of distributed generators in such cases is a complex task, as the performance depends on both the grid and control design. In this paper, we consider such a case where the optimal utilization of distributed generators is achieved by optimal power sharing while taking into account the grid topology, the available generators, and the way they are controlled. For the latter, we consider a droop-based decentralized control scheme whose primary objective is to achieve voltage regulation in the allowable operating range. A novel mixed-integer optimization approach is proposed to identify the optimal converter size and location in the network so that the microgrid can operate safely and with optimal use of the available resources. Time-domain simulations are used to validate the proposed approach and demonstrate its robustness to uncertainty in generator availability.

Sparse Poisson regression via mixed-integer optimization.

We present a mixed-integer optimization (MIO) approach to sparse Poisson regression. The MIO approach to sparse linear regression was first proposed in the 1970s, but has recently received renewed attention due to advances in optimization algorithms and computer hardware. In contrast to many sparse estimation algorithms, the MIO approach has the advantage of finding the best subset of explanatory variables with respect to various criterion functions. In this paper, we focus on a sparse Poisson regression that maximizes the weighted sum of the log-likelihood function and the L2-regularization term. For this problem, we derive a mixed-integer quadratic optimization (MIQO) formulation by applying a piecewise-linear approximation to the log-likelihood function. Optimization software can solve this MIQO problem to optimality. Moreover, we propose two methods for selecting a limited number of tangent lines effective for piecewise-linear approximations. We assess the efficacy of our method through computational experiments using synthetic and real-world datasets. Our methods provide better log-likelihood values than do conventional greedy algorithms in selecting tangent lines. In addition, our MIQO formulation delivers better out-of-sample prediction performance than do forward stepwise selection and L1-regularized estimation, especially in low-noise situations.

Optimal Claiming of Social Security Benefits

Using a lifecycle framework with Epstein-Zin (1989) utility and a mixed-integer optimization approach, we compute the optimal age to claim Social Security benefits. Taking advantage of homogeneity, a sufficient statistic is the ratio of wealth to the primary insurance amount (PIA). If the investor's wealth to PIA ratio exceeds a certain threshold, individuals should defer Social Security for at least a year. The optimal threshold depends on mortality assumptions and an individual’s utility preferences, but is less sensitive to capital market assumptions. The threshold wealth to PIA ratio increases from 5.5 for men and 5.2 for women at age 62 to 11.1 for men and 10.4 for women at age 69. Below the threshold wealth to PIA ratio, individuals claim Social Security to raise consumption. Above this level, investors can afford to fund consumption out of wealth for at least one year, and then claim a higher benefit.

Novel mixed integer optimization sparse regression approach in chemometrics

Sparse mathematical modelling plays an increasingly important role in chemometrics due to its interpretability and prediction power. While many sparse techniques used in chemometrics rely on L1 penalization to create sparser models, Mixed Integer Optimization (MIO) achieves sparsity by imposing constraints directly in the model. In this paper, we develop an intuitive and flexible robust sparse regression framework using MIO. We use constraints and penalization to achieve sparsity and robustness respectively. We test and compare results with those obtained using other techniques generating sparser models such as LASSO and sparse PLS. We also use PLS as a baseline to compare predictive performance. We use a LIBS data set of certified reference materials (CRM) of various mineral ores to illustrate the framework using different objective functions. The MIO framework proposed improves accuracy, sparsity and robustness vs. LASSO and SPLS. MIO achieves an average R2 higher than other methods on average by at least 10.6%. Robust MIO approach also improves interpretability. It also uses 4.3 variables on average while LASSO and SPLS use 16.1 and 805.8 respectively. We also illustrate how interpretability can help build better models through examples derived from the data sets used. When adding noise to the signal, MIO achieves an R2 of 0.69 on average when all models have negative R2 values. The MIO framework proposed is versatile and could be used in other chemometrics applications.

A mixed integer optimization approach for model selection in screening experiments

After completing the experimental runs of a screening design, the responses under study are analyzed by statistical methods to detect the active effects. To increase the chances of correctly identifying these effects, a good analysis method should provide alternative interpretations of the data, reveal the aliasing present in the design, and search only meaningful sets of effects as defined by user-specified restrictions such as effect heredity. This article presents a mixed integer optimization strategy to analyze data from screening designs that possesses all these properties. We illustrate our method by analyzing data from real and synthetic experiments, and using simulations.

Best subset selection via cross-validation criterion

This paper is concerned with the cross-validation criterion for selecting the best subset of explanatory variables in a linear regression model. In contrast with the use of statistical criteria (e.g., Mallows’ $$C_p$$, the Akaike information criterion, and the Bayesian information criterion), cross-validation requires only mild assumptions, namely, that samples are identically distributed and that training and validation samples are independent. For this reason, the cross-validation criterion is expected to work well in most situations involving predictive methods. The purpose of this paper is to establish a mixed-integer optimization approach to selecting the best subset of explanatory variables via the cross-validation criterion. This subset-selection problem can be formulated as a bilevel MIO problem. We then reduce it to a single-level mixed-integer quadratic optimization problem, which can be solved exactly by using optimization software. The efficacy of our method is evaluated through simulation experiments by comparison with statistical-criterion-based exhaustive search algorithms and $$L_1$$-regularized regression. Our simulation results demonstrate that, when the signal-to-noise ratio was low, our method delivered good accuracy for both subset selection and prediction.

Mixed-integer optimization approach to learning association rules for unplanned ICU transfer.

After admission to emergency department (ED), patients with critical illnesses are transferred to intensive care unit (ICU) due to unexpected clinical deterioration occurrence. Identifying such unplanned ICU transfers is urgently needed for medical physicians to achieve two-fold goals: improving critical care quality and preventing mortality. A priority task is to understand the crucial rationale behind diagnosis results of individual patients during stay in ED, which helps prepare for an early transfer to ICU. Most existing prediction studies were based on univariate analysis or multiple logistic regression to provide one-size-fit-all results. However, patient condition varying from case to case may not be accurately examined by such a simplistic judgment. In this study, we present a new decision tool using a mathematical optimization approach aiming to automatically discover rules associating diagnostic features with high-risk outcome (i.e., unplanned transfers) in different deterioration scenarios. We consider four mutually exclusive patient subgroups based on the principal reasons of ED visits: infections, cardiovascular/respiratory diseases, gastrointestinal diseases, and neurological/other diseases at a suburban teaching hospital. The analysis results demonstrate significant rules associated with unplanned transfer outcome for each subgroups and also show comparable prediction accuracy (>70%) compared to state-of-the-art machine learning methods while providing easy-to-interpret symptom-outcome information.

Learning a Mixture of Gaussians via Mixed-Integer Optimization

We consider the problem of estimating the parameters of a multivariate Gaussian mixture model (GMM) given access to n samples that are believed to have come from a mixture of multiple subpopulations. State-of-the-art algorithms used to recover these parameters use heuristics to either maximize the log-likelihood of the sample or try to fit first few moments of the GMM to the sample moments. In contrast, we present here a novel mixed-integer optimization (MIO) formulation that optimally recovers the parameters of the GMM by minimizing a discrepancy measure (either the Kolmogorov–Smirnov or the total variation distance) between the empirical distribution function and the distribution function of the GMM whenever the mixture component weights are known. We also present an algorithm for multidimensional data that optimally recovers corresponding means and covariance matrices. We show that the MIO approaches are practically solvable for data sets with n in the tens of thousands in minutes and achieve an average improvement of 60%–70% and 50%–60% on mean absolute percentage error in estimating the means and the covariance matrices, respectively, over the expectation–maximization (EM) algorithm independent of the sample size n. As the separation of the Gaussians decreases and, correspondingly, the problem becomes more difficult, the edge in performance in favor of the MIO methods widens. Finally, we also show that the MIO methods outperform the EM algorithm with an average improvement of 4%–5% on the out-of-sample accuracy for real-world data sets.

Mixed integer quadratic optimization formulations for eliminating multicollinearity based on variance inflation factor

Multicollinearity exists when some explanatory variables of a multiple linear regression model are highly correlated. High correlation among explanatory variables reduces the reliability of the analysis. To eliminate multicollinearity from a linear regression model, we consider how to select a subset of significant variables by means of the variance inflation factor (VIF), which is the most common indicator used in detecting multicollinearity. In particular, we adopt the mixed integer optimization (MIO) approach to subset selection. The MIO approach was proposed in the 1970s, and recently it has received renewed attention due to advances in algorithms and hardware. However, none of the existing studies have developed a computationally tractable MIO formulation for eliminating multicollinearity on the basis of VIF. In this paper, we propose mixed integer quadratic optimization (MIQO) formulations for selecting the best subset of explanatory variables subject to the upper bounds on the VIFs of selected variables. Our two MIQO formulations are based on the two equivalent definitions of VIF. Computational results illustrate the effectiveness of our MIQO formulations by comparison with conventional local search algorithms and MIO-based cutting plane algorithms.

A mixed-integer optimization approach for homogeneous magnet design

Magnetic resonance imaging (MRI) is a powerful diagnostic tool that has become the imaging modality of choice for soft-tissue visualization in radiation therapy. Emerging technologies aim to integrate MRI with a medical linear accelerator to form novel cancer therapy systems (MR-linac), but the design of these systems to date relies on heuristic procedures. This paper develops an exact, optimization-based approach for magnet design that 1) incorporates the most accurate physics calculations to date, 2) determines precisely the relative spatial location, size, and current magnitude of the magnetic coils, 3) guarantees field homogeneity inside the imaging volume, 4) produces configurations that satisfy, for the first time, small-footprint feasibility constraints required for MR-linacs. Our approach leverages modern mixed-integer programming (MIP), enabling significant flexibility in magnet design generation, e.g., controlling the number of coils and enforcing symmetry between magnet poles. Our numerical results demonstrate the superiority of our method versus current mainstream methods.

Mixed integer optimization approach to groundwater pollution source identification problems

ABSTRACTIn this study, we consider an approach that employs 0–1 type integer programming in a simulation-optimization model to identify the characteristics of groundwater pollutant sources. The advantage of the proposed solution model is that it reduces the dimensions of the unknown variable, where 0–1 type integer variables denote the source position. If the estimated location variable is equal to 0, this means that the corresponding location is not the actual source location and there is no need to estimate its source flux. If the estimated location variable is equal to 1, this means that the corresponding location is the actual source location and the source flux needs to be identified. We evaluated the performance of the proposed model based on four scenarios with different potential source location sets, incomplete data conditions, and noisy data conditions. The results obtained indicated that the proposed solution model is an effective approach for solving groundwater pollution source identification problems.

Best subset selection via a modern optimization lens

In the period 1991–2015, algorithmic advances in Mixed Integer Optimization (MIO) coupled with hardware improvements have resulted in an astonishing 450 billion factor speedup in solving MIO problems. We present a MIO approach for solving the classical best subset selection problem of choosing $k$ out of $p$ features in linear regression given $n$ observations. We develop a discrete extension of modern first-order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with $n$ in the 1000s and $p$ in the 100s in minutes to provable optimality, and finds near optimal solutions for $n$ in the 100s and $p$ in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than Lasso and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.

Optimizing over coherent risk measures and non-convexities: a robust mixed integer optimization approach

Recently, coherent risk measure minimization was formulated as robust optimization and the correspondence between coherent risk measures and uncertainty sets of robust optimization was investigated. We study minimizing coherent risk measures under a norm equality constraint with the use of robust optimization formulation. Not only existing coherent risk measures but also a new coherent risk measure is investigated by setting a new uncertainty set. The norm equality constraint itself has a practical meaning or plays a role to prevent a meaningless solution, the zero vector, in the context of portfolio optimization or binary classification in machine learning, respectively. For such advantages, the convexity is sacrificed in the formulation. However, we show a condition for an input of our problem which guarantees that the nonconvex constraint is convexified without changing the optimality of the problem. If the input does not satisfy the condition, we propose to solve a mixed integer optimization problem by using the $$\ell _1$$ or $$\ell _\infty $$ -norm. The numerical experiments show that our approach has good performance for portfolio optimization and binary classification and also imply its flexibility of modelling that makes it possible to deal with various coherent risk measures.

Least quantile regression via modern optimization

We address the Least Quantile of Squares (LQS) (and in particular the Least Median of Squares) regression problem using modern optimization methods. We propose a Mixed Integer Optimization (MIO) formulation of the LQS problem which allows us to find a provably global optimal solution for the LQS problem. Our MIO framework has the appealing characteristic that if we terminate the algorithm early, we obtain a solution with a guarantee on its sub-optimality. We also propose continuous optimization methods based on first-order subdifferential methods, sequential linear optimization and hybrid combinations of them to obtain near optimal solutions to the LQS problem. The MIO algorithm is found to benefit significantly from high quality solutions delivered by our continuous optimization based methods. We further show that the MIO approach leads to (a) an optimal solution for any dataset, where the data-points $(y_i,\mathbf{x}_i)$'s are not necessarily in general position, (b) a simple proof of the breakdown point of the LQS objective value that holds for any dataset and (c) an extension to situations where there are polyhedral constraints on the regression coefficient vector. We report computational results with both synthetic and real-world datasets showing that the MIO algorithm with warm starts from the continuous optimization methods solve small ($n=100$) and medium ($n=500$) size problems to provable optimality in under two hours, and outperform all publicly available methods for large-scale ($n={}$10,000) LQS problems.

Managing daily surgery schedules in a teaching hospital: a mixed-integer optimization approach.

BackgroundThis study examined the daily surgical scheduling problem in a teaching hospital. This problem relates to the use of multiple operating rooms and different types of surgeons in a typical surgical day with deterministic operation durations (preincision, incision, and postincision times). Teaching hospitals play a key role in the health-care system; however, existing models assume that the duration of surgery is independent of the surgeon’s skills. This problem has not been properly addressed in other studies. We analyze the case of a Spanish public hospital, in which continuous pressures and budgeting reductions entail the more efficient use of resources.MethodsTo obtain an optimal solution for this problem, we developed a mixed-integer programming model and user-friendly interface that facilitate the scheduling of planned operations for the following surgical day. We also implemented a simulation model to assist the evaluation of different dispatching policies for surgeries and surgeons. The typical aspects we took into account were the type of surgeon, potential overtime, idling time of surgeons, and the use of operating rooms.ResultsIt is necessary to consider the expertise of a given surgeon when formulating a schedule: such skill can decrease the probability of delays that could affect subsequent surgeries or cause cancellation of the final surgery. We obtained optimal solutions for a set of given instances, which we obtained through surgical information related to acceptable times collected from a Spanish public hospital.ConclusionsWe developed a computer-aided framework with a user-friendly interface for use by a surgical manager that presents a 3-D simulation of the problem. Additionally, we obtained an efficient formulation for this complex problem. However, the spread of this kind of operation research in Spanish public health hospitals will take a long time since there is a lack of knowledge of the beneficial techniques and possibilities that operational research can offer for the health-care system.Electronic supplementary materialThe online version of this article (doi:10.1186/1472-6963-14-464) contains supplementary material, which is available to authorized users.

Optimization of AC Transmission System Planning

This paper presents a mixed-integer optimization approach for transmission system planning with an AC network model. The approach makes use of the conic programming relaxation of the optimal power flow problem to define a disjunctive program that takes into account both the network power flow and the voltage limits. The proposed mixed-integer optimization program has a convex relaxation and can be solved using a branch-and-cut algorithm. Planning results show that the method produces solutions that satisfy the AC network model constraints without further reinforcement and are at less cost as compared to solutions reported in the literature.

A mixed integer optimisation approach for integrated water resources management

In areas lacking substantial freshwater resources, the utilisation of alternative water sources, such as desalinated seawater and reclaimed water, is a sustainable alternative option. This paper presents an optimisation approach for the integrated management of water resources, including desalinated seawater, wastewater and reclaimed water, for insular water deficient areas. The proposed mixed integer linear programming (MILP) model takes into account the subdivided regions on the island, the subsequent localised needs for water use (including water quality) and wastewater production, as well as geographical aspects. In addition, the integration of potable and non-potable water systems is considered. The optimal water management decisions, including the location of desalination, wastewater treatment, and reclamation plants, as well as the conveyance infrastructure for desalinated water, wastewater and reclaimed water, are obtained by minimising the annualised total capital and operating costs. Finally, the proposed approach is applied to two Greek islands: Syros and Paros-Antiparos, for case study and scenario analysis.

Semiconductor lot allocation using robust optimization

In this work, the problem of allocating a set of production lots to satisfy customer orders is considered. This research is of relevance to lot-to-order matching problems in semiconductor supply chain settings. We consider that lot-splitting is not allowed during the allocation process due to standard practices. Furthermore, lot-sizes are regarded as uncertain planning data when making the allocation decisions due to potential yield loss. In order to minimize the total penalties of demand un-fulfillment and over-fulfillment, a robust mixed-integer optimization approach is adopted to model is proposed the problem of allocating a set of work-in-process lots to customer orders, where lot-sizes are modeled using ellipsoidal uncertainty sets. To solve the optimization problem efficiently we apply the techniques of branch-and-price and Benders decomposition. The advantages of our model are that it can represent uncertainty in a straightforward manner with little distributional assumptions, and it can produce solutions that effectively hedge against the uncertainty in the lot-sizes using very reasonable amounts of computational effort.

Improved Mixed Integer Optimization Approach for Data Rectification with Gross Error Candidates

Mixed integer linear programming (MILP) approach for simultaneous gross error detection and data reconciliation has been proved as an efficient way to adjust process data with material, energy, and other balance constrains. But the efficiency will decrease significantly when this method is applied in a large-scale problem because there are too many binary variables involved. In this article, an improved method is proposed in order to generate gross error candidates with reliability factors before data rectification. Candidates are used in the MILP objective function to improve the efficiency and accuracy by reducing the number of binary variables and giving accurate weights for suspected gross errors candidates. Performance of this improved method is compared and discussed by applying the algorithm in a widely used industrial example.

A mixed-integer optimization approach for polygeneration energy systems design

A mixed-integer nonlinear programming (MINLP) model is developed for the design optimization of polygeneration energy systems. A suitable superstructure is introduced, based on partitioning a general polygeneration energy system into four major blocks, for each of which alternative available technologies and types of equipment are considered. A detailed case study, involving a coal-based polygeneration plant producing electricity and methanol, is presented to demonstrate the key features and applicability of the proposed approach.