EURO Journal on Computational Optimization最新文献

This paper presents exact Semi-Definite Program (SDP) reformulations for infinite-dimensional moment optimization problems involving a new class of piecewise Sum-of-Squares (SOS)-convex functions and projected spectrahedral support sets. These reformulations show that solving a single SDP finds the optimal value and an optimal probability measure of the original moment problem. This is done by establishing an SOS representation for the non-negativity of a piecewise SOS-convex function over a projected spectrahedron. Finally, as an application and a proof-of-concept illustration, the paper presents numerical results for the Newsvendor and revenue maximization problems with higher-order moments by solving their equivalent SDP reformulations. These reformulations promise a flexible and efficient approach to solving these models. The main novelty of the present work in relation to the recent research lies in finding the solution to moment problems, for the first time, with piecewise SOS-convex functions from their numerically tractable exact SDP reformulations.

The interpretability of models has become a crucial issue in Machine Learning because of algorithmic decisions' growing impact on real-world applications. Tree ensemble methods, such as Random Forests or XgBoost, are powerful learning tools for classification tasks. However, while combining multiple trees may provide higher prediction quality than a single one, it sacrifices the interpretability property resulting in “black-box” models. In light of this, we aim to develop an interpretable representation of a tree-ensemble model that can provide valuable insights into its behavior. First, given a target tree-ensemble model, we develop a hierarchical visualization tool based on a heatmap representation of the forest's feature use, considering the frequency of a feature and the level at which it is selected as an indicator of importance. Next, we propose a mixed-integer linear programming (MILP) formulation for constructing a single optimal multivariate tree that accurately mimics the target model predictions. The goal is to provide an interpretable surrogate model based on oblique hyperplane splits, which uses only the most relevant features according to the defined forest's importance indicators. The MILP model includes a penalty on feature selection based on their frequency in the forest to further induce sparsity of the splits. The natural formulation has been strengthened to improve the computational performance of mixed-integer software. Computational experience is carried out on benchmark datasets from the UCI repository using a state-of-the-art off-the-shelf solver. Results show that the proposed model is effective in yielding a shallow interpretable tree approximating the tree-ensemble decision function.

Classifications organize entities into categories that identify similarities within a category and discern dissimilarities among categories, and they powerfully classify information in support of analysis. We propose a new classification scheme premised on the reality of imperfect data. Our computational model uses uncertain data envelopment analysis to define a classification's proximity to equitable efficiency, which is an aggregate measure of intra-similarity within a classification's categories. Our classification process has two overriding computational challenges, those being a loss of convexity and a combinatorially explosive search space. We overcome the first challenge by establishing lower and upper bounds on the proximity value, and then by searching this range with a first-order algorithm. We address the second challenge by adapting the p-median problem to initiate our exploration, and by then employing an iterative neighborhood search to finalize a classification. We conclude by classifying the thirty stocks in the Dow Jones Industrial average into performant tiers, by classifying prostate treatments into clinically effectual categories, and dividing airlines into peer groups.

Time series shapelets are a state-of-the-art data mining technique that is applied to time series supervised classification tasks. Shapelets are defined as subsequences that retain the most discriminating power contained in time series. The main advantage of shapelets-based methods consists of their great interpretability. Indeed, shapelets can provide the end-user with very helpful insights about the most interesting subsequences. In this paper, we propose a novel Mixed-Integer Programming model to optimize shapelets discovery based on optimal binary decision trees. Our formulation provides a flexible and adaptable classification framework that is interpretable with respect to both the mathematical model and the final output. Computational results for a large class of datasets show that our approach achieves performance comparable with state-of-the-art shapelets-based classification methods. Our model is the first approach based on optimal decision tree induction for time series classification.

In civil aircraft, two partially redundant hydraulic circuits typically power various systems. During assembly, a critical phase involves simultaneously rinsing and purging these hydraulic circuits using loops. Precedence constraints are necessary to prevent the recontamination of already rinsed loops, leading to increased rinsing time. This paper presents this problem as a unique instance of the Resource Constrained Parallel Machine Scheduling Problem, where each circuit represents a machine, pipe loops to be rinsed represent jobs, and machines share a hydraulic power source. For two dedicated processors and a single resource, an optimal schedule minimizing the makespan can be generated in polynomial time. However, due to the requirement of rinsing certain pipe loops on a circuit before others, there are precedence constraints between some jobs within the same circuit. By employing a reduction of the 3-partition problem, we demonstrate that this situation results in a problem that is NP-hard in the strong sense. We evaluate several Mixed-Integer Linear Programming and Constraint Programming formulations of the problem, using Cplex, CPO, Gurobi, and Z3, against several proposed heuristics. Given that the size of the instances we need to solve exceeds what can be solved in acceptable time by solvers, we propose a heuristic and compare its performance with the optimum.

Due to the continued success of machine learning and deep learning in particular, supervised classification problems are ubiquitous in numerous scientific fields. Training these models typically involves the minimization of the empirical risk over large data sets along with a possibly non-differentiable regularization. In this paper, we introduce a stochastic gradient method for the considered classification problem. To control the variance of the objective's gradients, we use an automatic sample size selection along with a variable metric to precondition the stochastic gradient directions. Further, we utilize a non-monotone line search to automatize step size selection. Convergence results are provided for both convex and non-convex objective functions. Extensive numerical experiments verify that the suggested approach performs on par with state-of-the-art methods for training both statistical models for binary classification and artificial neural networks for multi-class image classification. The code is publicly available at https://github.com/koblererich/lisavm.

We develop a Design and Analysis of the Computer Experiments (DACE) approach to the stochastic unit commitment problem for power systems with significant renewable integration. For this purpose, we use a two-stage stochastic programming formulation of the stochastic unit commitment-economic dispatch problem. Typically, a sample average approximation of the true problem is solved using a cutting plane method (such as the L-shaped method) or scenario decomposition (such as Progressive Hedging) algorithms. However, when the number of scenarios increases, these solution methods become computationally prohibitive. To address this challenge, we develop a novel DACE approach that exploits the structure of the first-stage unit commitment decision space in a design of experiments, uses features based upon solar generation, and trains a multivariate adaptive regression splines model to approximate the second stage of the stochastic unit commitment-economic dispatch problem. We conduct experiments on two modified IEEE-57 and IEEE-118 test systems and assess the quality of the solutions obtained from both the DACE and the L-shaped methods in a replicated procedure. The results obtained from this approach attest to the significant improvement in the computational performance of the DACE approach over the traditional L-shaped method.

We introduce a heuristic rule for calculating the stepsize in the subgradient method for unconstrained convex nonsmooth optimization which, unlike the classic approach, is based on retaining some information from previous iteration. The rule is inspired by the well known two-point stepsize by Barzilai and Borwein (BB) [6] for smooth optimization and it coincides with (BB) in case the function to be minimised is convex quadratic.

Under the use of appropriate safeguards we demonstrate that the method terminates at a point that satisfies an approximate optimality condition.

The proposed approach is tested in the framework of Lagrangian relaxation for integer linear programming where the Lagrangian dual requires maximization of a concave and nonsmooth (piecewise affine) function. In particular we focus on the relaxation of the Minimum Spanning Tree problem with Conflicting Edge Pairs (MSTC). Comparison with classic subgradient method is presented. The results on some widely used academic test problems are provided too.