Discrete optimization

About: Discrete optimization is a research topic. Over the lifetime, 4598 publications have been published within this topic receiving 158297 citations. The topic is also known as: discrete optimisation.

Toward multidimensional assignment data association in robot localization and mapping

Abstract: It is well accepted that the data association or the correspondence problem is one of the toughest problems faced by any state estimation algorithm. Particularly in robotics, it is not very well addressed. This paper introduces a multidimensional assignment (MDA)-based data association algorithm for the simultaneous localization and map building (SLAM) problem in mobile robot navigation. The data association problem is cast in a general discrete optimization framework and the MDA formulation for multitarget tracking is extended for SLAM using sensor location uncertainty with the joint likelihood of measurements over multiple frames as the objective function. Methods for feature initialization and management are also integrated into the algorithm. When clutter is high and features are sparse, the compatibility information of features of a single measurement frame is not sufficient to make effective data-association decisions,thus compromising performance of single-frame-based methods. However, in a multiple-measurement-frame approach, the availability of more than one frame of measurement provides for more effective data-association decisions to be made, as consistency of measurements are looked at in several frames of measurement. Simulations are conducted to verify the performance gains over the conventional nearest neighbor (NN) data association algorithm and the joint compatibility branch and bound (JCBB) algorithm, especially in the presence of varying densities of spurious measurements and dynamic objects. Experimental results with ground truth are presented to demonstrate the practicality of the proposed data-association method in complex and large outdoor environments and its effectiveness over single-frame-based NN and JCBB schemes.

Bayesian Optimization of Combinatorial Structures

Abstract: The optimization of expensive-to-evaluate black-box functions over combinatorial structures is an ubiquitous task in machine learning, engineering and the natural sciences. The combinatorial explosion of the search space and costly evaluations pose challenges for current techniques in discrete optimization and machine learning, and critically require new algorithmic ideas. This article proposes, to the best of our knowledge, the first algorithm to overcome these challenges, based on an adaptive, scalable model that identifies useful combinatorial structure even when data is scarce. Our acquisition function pioneers the use of semidefinite programming to achieve efficiency and scalability. Experimental evaluations demonstrate that this algorithm consistently outperforms other methods from combinatorial and Bayesian optimization.

Typical Properties of Winners and Losers [0.2ex] in Discrete Optimization

Abstract: We present a probabilistic analysis of a large class of combinatorial optimization problems containing all binary optimization problems defined by linear constraints and a linear objective function over $\{0,1\}^n$. Our analysis is based on a semirandom input model that preserves the combinatorial structure of the underlying optimization problem by parameterizing which input numbers are of a stochastic and which are of an adversarial nature. This input model covers various probability distributions for the choice of the stochastic numbers and includes smoothed analysis with Gaussian and other kinds of perturbation models as a special case. In fact, we can exactly characterize the smoothed complexity of binary optimization problems in terms of their worst-case complexity: A binary optimization problem has polynomial smoothed complexity if and only if it admits a (possibly randomized) algorithm with pseudo-polynomial worst-case complexity. Our analysis is centered around structural properties of binary optimization problems, called winner, loser, and feasibility gap. We show that if the coefficients of the objective function are stochastic, then the gap between the best and second best solution is likely to be of order $\Omega(1/n)$. Furthermore, we show that if the coefficients of the constraints are stochastic, then the slack of the optimal solution with respect to this constraint is typically of order $\Omega(1/n^2)$. We exploit these properties in an adaptive rounding scheme that increases the accuracy of calculation until the optimal solution is found. The strength of our techniques is illustrated by applications to various pc-hard optimization problems from mathematical programming, network design, and scheduling for which we obtain the first algorithms with polynomial smoothed/average-case complexity.

A theory of lexicographic multi-criteria optimization

Abstract: The field of multi-criteria optimization is reviewed as it pertains to lexicographic optimization over real-valued vector spaces. How lexicographic optimization differs from multi-criteria optimization that is restricted to proper Pareto optima is explained. Through a survey of previous work, it is revealed that there are currently no generally applicable methods for solving lexicographic optimization problems, and it is explained that this is due to the lack of an adequate mathematical theory for such problems. A more adequate mathematical theory is then presented for lexicographic optimization in this paper.

Parallel branch, cut, and price for large-scale discrete optimization

Abstract: In discrete optimization, most exact solution approaches are based on branch and bound, which is conceptually easy to parallelize in its simplest forms. More sophisticated variants, such as the so-called branch, cut, and price algorithms, are more difficult to parallelize because of the need to share large amounts of knowledge discovered during the search process. In the first part of the paper, we survey the issues involved in parallelizing such algorithms. We then review the implementation of SYMPHONY and COIN/BCP, two existing frameworks for implementing parallel branch, cut, and price. These frameworks have limited scalability, but are effective on small numbers of processors. Finally, we briefly describe our next-generation framework, which improves scalability and further abstracts many of the notions inherent in parallel BCP, making it possible to implement and parallelize more general classes of algorithms.