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Register a new userInterior Point Solving for LP-based prediction+optimisation
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Jayanta Mandi
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Solving optimization problems is the key to decision making in many real-life analytics applications. However, the coefficients of the optimization problems are often uncertain and dependent on external factors, such as future demand or energy or stock prices. Machine learning (ML) models, especially neural networks, are increasingly being used to estimate these coefficients in a data-driven way. Hence, end-to-end predict-and-optimize approaches, which consider how effective the predicted values are to solve the optimization problem, have received increasing attention. In case of integer linear programming problems, a popular approach to overcome their non-differentiabilty is to add a quadratic penalty term to the continuous relaxation, such that results from differentiating over quadratic programs can be used. Instead we investigate the use of the more principled logarithmic barrier term, as widely used in interior point solvers for linear programming. Specifically, instead of differentiating the KKT conditions, we consider the homogeneous self-dual formulation of the LP and we show the relation between the interior point step direction and corresponding gradients needed for learning. Finally our empirical experiments demonstrate our approach performs as good as if not better than the state-of-the-art QPTL (Quadratic Programming task loss) formulation of Wilder et al. and SPO approach of Elmachtoub and Grigas.
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The main feature of large-scale multi-objective optimization problems (LSMOP) is to optimize multiple conflicting objectives while considering thousands of decision variables at the same time. An efficient LSMOP algorithm should have the ability to escape the local optimal solution from the huge search space and find the global optimal. Most of the current researches focus on how to deal with decision variables. However, due to the large number of decision variables, it is easy to lead to high computational cost. Maintaining the diversity of the population is one of the effective ways to improve search efficiency. In this paper, we propose a probabilistic prediction model based on trend prediction model and generating-filtering strategy, called LT-PPM, to tackle the LSMOP. The proposed method enhances the diversity of the population through importance sampling. At the same time, due to the adoption of an individual-based evolution mechanism, the computational cost of the proposed method is independent of the number of decision variables, thus avoiding the problem of exponential growth of the search space. We compared the proposed algorithm with several state-of-the-art algorithms for different benchmark functions. The experimental results and complexity analysis have demonstrated that the proposed algorithm has significant improvement in terms of its performance and computational efficiency in large-scale multi-objective optimization.
We apply novel inner-iteration preconditioned Krylov subspace methods to the interior-point algorithm for linear programming (LP). Inner-iteration preconditioners recently proposed by Morikuni and Hayami enable us to overcome the severe ill-conditioning of linear equations solved in the final phase of interior-point iterations. The Krylov subspace methods do not suffer from rank-deficiency and therefore no preprocessing is necessary even if rows of the constraint matrix are not linearly independent. By means of these methods, a new interior-point recurrence is proposed in order to omit one matrix-vector product at each step. Extensive numerical experiments are conducted over diverse instances of 138 LP problems including the Netlib, QAPLIB, Mittelmann and Atomizer Basis Pursuit collections. The largest problem has 434,580 unknowns. It turns out that our implementation is more robust than the standard public domain solvers SeDuMi (Self-Dual Minimization), SDPT3 (Semidefinite Programming Toh-Todd-Tutuncu) and the LSMR iterative solver in PDCO (Primal-Dual Barrier Method for Convex Objectives) without increasing CPU time. The proposed interior-point method based on iterative solvers succeeds in solving a fairly large number of LP instances from benchmark libraries under the standard stopping criteria. The work also presents a fairly extensive benchmark test for several renowned solvers including direct and iterative solvers.
Population-based evolutionary algorithms have great potential to handle multiobjective optimisation problems. However, these algorithms depends largely on problem characteristics, and there is a need to improve their performance for a wider range of problems. References, which are often specified by the decision makers preference in different forms, are a very effective method to improve the performance of algorithms but have not been fully explored in literature. This paper proposes a novel framework for effective use of references to strengthen algorithms. This framework considers references as search targets which can be adjusted based on the information collected during the search. The proposed framework is combined with new strategies, such as reference adaptation and adaptive local mating, to solve different types of problems. The proposed algorithm is compared with state of the arts on a wide range of problems with diverse characteristics. The comparison and extensive sensitivity analysis demonstrate that the proposed algorithm is competitive and robust across different types of problems studied in this paper.
Quality-Diversity algorithms refer to a class of evolutionary algorithms designed to find a collection of diverse and high-performing solutions to a given problem. In robotics, such algorithms can be used for generating a collection of controllers covering most of the possible behaviours of a robot. To do so, these algorithms associate a behavioural descriptor to each of these behaviours. Each behavioural descriptor is used for estimating the novelty of one behaviour compared to the others. In most existing algorithms, the behavioural descriptor needs to be hand-coded, thus requiring prior knowledge about the task to solve. In this paper, we introduce: Autonomous Robots Realising their Abilities, an algorithm that uses a dimensionality reduction technique to automatically learn behavioural descriptors based on raw sensory data. The performance of this algorithm is assessed on three robotic tasks in simulation. The experimental results show that it performs similarly to traditional hand-coded approaches without the requirement to provide any hand-coded behavioural descriptor. In the collection of diverse and high-performing solutions, it also manages to find behaviours that are novel with respect to more features than its hand-coded baselines. Finally, we introduce a variant of the algorithm which is robust to the dimensionality of the behavioural descriptor space.
This study suggests a new approach to EEG data classification by exploring the idea of using evolutionary computation to both select useful discriminative EEG features and optimise the topology of Artificial Neural Networks. An evolutionary algorithm is applied to select the most informative features from an initial set of 2550 EEG statistical features. Optimisation of a Multilayer Perceptron (MLP) is performed with an evolutionary approach before classification to estimate the best hyperparameters of the network. Deep learning and tuning with Long Short-Term Memory (LSTM) are also explored, and Adaptive Boosting of the two types of models is tested for each problem. Three experiments are provided for comparison using different classifiers: one for attention state classification, one for emotional sentiment classification, and a third experiment in which the goal is to guess the number a subject is thinking of. The obtained results show that an Adaptive Boosted LSTM can achieve an accuracy of 84.44%, 97.06%, and 9.94% on the attentional, emotional, and number datasets, respectively. An evolutionary-optimised MLP achieves results close to the Adaptive Boosted LSTM for the two first experiments and significantly higher for the number-guessing experiment with an Adaptive Boosted DEvo MLP reaching 31.35%, while being significantly quicker to train and classify. In particular, the accuracy of the nonboosted DEvo MLP was of 79.81%, 96.11%, and 27.07% in the same benchmarks. Two datasets for the experiments were gathered using a Muse EEG headband with four electrodes corresponding to TP9, AF7, AF8, and TP10 locations of the international EEG placement standard. The EEG MindBigData digits dataset was gathered from the TP9, FP1, FP2, and TP10 locations.
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