Mixed-integer Algorithm Research Articles

Fast mixed integer optimization (FMIO) for high dose rate brachytherapy

The purpose of this work was to develop an efficient quadratic mixed integer programming algorithm for high dose rate (HDR) brachytherapy treatment planning problems and integrate the algorithm into an open-source Monte Carlo based treatment planning software, RapidBrachyMCTPS. The mixed-integer algorithm yields a globally optimum solution to the dose volume histogram (DVH) based problem and, unlike other methods, is not susceptible to local minimum trapping. A hybrid linear-quadratic penalty model coupled to a mixed integer programming model was used to optimize treatment plans for 10 prostate cancer patients. Dose distributions for each dwell position were calculated with RapidBrachyMCTPS with type A uncertainties less than 0.2% in voxels within the planning target volume (PTV). The optimization process was divided into two parts. First, the data was preprocessed, in which the problem size was reduced by eliminating voxels that had negligible impact on the solution (e.g. far from the dwell position). Second, the best combination of dwell times to obtain a plan with the highest score was found. The dwell positions and dose volume constraints were used as input to a commercial mixed integer optimizer (Gurobi Optimization, Inc.). A penalty-based criterion was adopted for the scoring. The voxel-reduction technique successfully reduced the problem size by an average of 91%, without loss of quality. The preprocessing of the optimization process required on average 4 s and solving for the global maximum required on average 33 s. The total optimization time averaged 37 s, which is a substantial improvement over the ∼15 min optimization time reported in published literature. The plan quality was evaluated by evaluating dose volume metrics, including PTV D 90, rectum and bladder D 1cc and urethra D 0.1cc . In conclusion, fast mixed integer optimization is an order of magnitude faster than current mixed-integer approaches for solving HDR brachytherapy treatment planning problems with DVH based metrics.

Mixed-Integer Algorithm for Optimal Dispatch of Integrated PV-Storage Systems

The exploitation of combined photovoltaic (PV) plants and storage systems is nowadays assuming growing importance, due to the technical, environmental, and economical benefits which can derive from an optimal integration. In this paper, a mixed-integer algorithm for the optimal dispatch of a storage system, based on the day-ahead PV forecasting is developed. The optimization objective is the maximization of the total production of the integrated system, according to a requested active power profile, which can be defined by the operator. The study case of an existing distribution management system, which operates on the low-voltage microgrid at University of Genova is analyzed. The procedure is validated by field results with particular attention to the storage round-trip efficiency.

A branch and price algorithm for a Stackelberg Security Game

Mixed integer optimization formulations are an attractive alternative to solve Stackelberg Game problems thanks to the efficiency of state of the art mixed integer algorithms. In particular, decomposition algorithms, such as branch and price methods, make it possible to tackle instances large enough to represent games inspired in real world domians.In this work we focus on Stackelberg Games that arise from a security application and investigate the use of a new branch and price method to solve its mixed integer optimization formulation. We prove that the algorithm provides upper and lower bounds on the optimal solution at every iteration and investigate the use of stabilization heuristics. Our preliminary computational results compare this solution approach with previous decomposition methods obtained from alternative integer programming formulations of Stackelberg games.

Efficient Hyperparameter Optimization for Deep Learning Algorithms Using Deterministic RBF Surrogates

Automatically searching for optimal hyperparameter configurations is of crucial importance for applying deep learning algorithms in practice. Recently, Bayesian optimization has been proposed for optimizing hyperparameters of various machine learning algorithms. Those methods adopt probabilistic surrogate models like Gaussian processes to approximate and minimize the validation error function of hyperparameter values. However, probabilistic surrogates require accurate estimates of sufficient statistics (e.g., covariance) of the error distribution and thus need many function evaluations with a sizeable number of hyperparameters. This makes them inefficient for optimizing hyperparameters of deep learning algorithms, which are highly expensive to evaluate. In this work, we propose a new deterministic and efficient hyperparameter optimization method that employs radial basis functions as error surrogates. The proposed mixed integer algorithm, called HORD, searches the surrogate for the most promising hyperparameter values through dynamic coordinate search and requires many fewer function evaluations. HORD does well in low dimensions but it is exceptionally better in higher dimensions. Extensive evaluations on MNIST and CIFAR-10 for four deep neural networks demonstrate HORD significantly outperforms the well-established Bayesian optimization methods such as GP, SMAC, and TPE. For instance, on average, HORD is more than 6 times faster than GP-EI in obtaining the best configuration of 19 hyperparameters.

Spending Scarce Funds More Efficiently—Including the Pattern of Interdependence in Cost-Benefit Analysis

The financial crisis has substantially aggravated shortages of public funds all over the world, particularly in many developed countries. After the expiration of investments made within the framework of public economic recovery plans, the availability of public funds for infrastructure investments will be subject to severe constraints. These conditions require that scarce available funds be spent in an efficient and effective manner. Evaluation methods to select and prioritize infrastructure projects of an investment package often presuppose that infrastructure projects are independent of one another. However, it is inherent in the nature of transport networks that links are interdependent and that changes in one link will affect other parts of the network. Some infrastructure projects might be characterized by substitutive interdependence, while others might interact in a synergistic context. Thus, the cost and benefits of a project are strongly dependent on the realization of other projects. This paper confronts this issue by comprehensively addressing the nature of the problem to derive an optimal set of investment projects. The approach utilizes a dynamic mixed integer algorithm, and the complexity of the combinatorial problem is reduced by elaborating on a heuristic method. Under application of these requisites, this paper develops a framework for integrating the interdependence between infrastructure projects in cost-benefit analysis. Results obtained from an example application related to investments made in the Trans-European Network (TEN-T) emphasize the importance and relevance of considering interdependence for evaluating the investment potential of infrastructure projects.

Study on optimization of operation for multipurpose batch chemical plants with multiple production routes

According to the operation characteristics of such plants, on the basis of Mauderli—Wellons's research, the mathematical models of operation optimization are modified which consider the campaign formation and production plan problem respectively. A new heuristics is advanced to search for the dominant production campaigns. Comparing the heuristics with Mauderli's algorithm, it ensures that parallel items in—phase have the same batch and that each production line operates with optimized batch, therefore the plant has a larger processing rate. This heuristics can effectively relieve the bottlenecks, and avoid the complexity resulting from the nonconvexity of the model. Within the resource alocated, the mixed—integer algorithm is used to maximize the profit.

The reliability of optimization under dose-volume limits

Purpose : An optimization algorithm improves the distribution of dose among discrete points in tissues, but tolerance depends on the distribution of dose across a continuous volume. This report asks whether an exact algorithm can be completed when enough points are taken to accurately model a dose-volume constraint. Methods and Materials : Trials were performed using a 3-dimensional model of conformal therapy of lung cancer Trial were repeated with different limits placed on the fraction of lung which could receive > 20 Gy. Bounds were placed on cord dose and target dose inhomogeneity. A mixed integer algorithm was used to find a feasible set of beam weights which would maximize tumor dose. Tests of feasibility and optimality are introduced to check the solution accuracy. Results : Solutions were optimal for points used to model tissues. An accuracy of 3–4% in a volume condition could be obtained with models of 450–600 points. The error improved to 2% with 800 points to model the lung. Solution times increased six-fold at this level of accuracy. Conclusion : The mixed integer method can find optimum weights which respect dose-volume conditions in usually acceptable times. If constraints are violated by an excessive amount, the optimization model should be rerun with more points.

Technical Note—An Improved Branch-and-Bound Method for Integer Programming

This note proposes two extensions of the successful Beale and Small branch-and-bound mixed-integer algorithm. The integer requirements on nonbasic variables are utilized to calculate stronger “penalties” when searching down the solution tree and to give a stronger criterion for abandoning unprofitable branches of the tree when backtracking. This stronger criterion is obtained by making use of Gomory cutting-plane constraints. These modifications have produced considerable reductions of the searching effort required for pure integer and predominantly integer problems, and have the further advantage of being very easy to incorporate.

Algorithms for the Simple Plant-Location Problem with Some Side Conditions

The algorithms of this paper belong to the direct-search or implicit-enumeration type. They compare to the recently proposed algorithm of Efroymson and Ray, as does the mixed integer algorithm proposed by C. E. Lemke and the author to that of Land and Doig. The general plan of procedure is expected to be equally valid for the capacitated plant-location problem and for transshipment problems with fixed charges; with some of the proposed devices more important for these difficult problems than for the simple plant location problem. Considerable computational experience has been accumulated and is discussed at some length. It suggests that additional work on the construction of “adaptive” programs, matching algorithm to data structure, is desirable.

An Extension of the Gomory Mixed-Integer Algorithm to Mixed-Discrete Variables

The methods of R. E. Gomory for the iterative solution of the mixed-integer linear programming problem are extended directly to the case where some or all of the variables are nonuniformly discrete, i.e., they are restricted to assume values from certain specified sets of unequally-spaced constants. The algorithm presented is shown to converge in a finite number of steps for a discrete-valued objective function.