The Voice of Optimization

We introduce the idea that using optimal classification trees (OCTs) and optimal classification trees with-hyperplanes (OCT-Hs), interpretable machine learning algorithms developed by Bertsimas and Dunn [2017, 2018], we are able to obtain insight on the strategy behind the optimal solution in continuous and mixed-integer convex optimization problem as a function of key parameters that affect the problem. In this way, optimization is not a black box anymore. Instead, we redefine optimization as a multiclass classification problem where the predictor gives insights on the logic behind the optimal solution. In other words, OCTs and OCT-Hs give optimization a voice. We show on several realistic examples that the accuracy behind our method is in the 90%-100% range, while even when the predictions are not correct, the degree of suboptimality or infeasibility is very low. We compare optimal strategy predictions of OCTs and OCT-Hs and feedforward neural networks (NNs) and conclude that the performance of OCT-Hs and NNs is comparable. OCTs are somewhat weaker but often competitive. Therefore, our approach provides a novel insightful understanding of optimal strategies to solve a broad class of continuous and mixed-integer optimization problems

Fast and Lean Immutable Multi-Maps on the JVM based on Heterogeneous Hash-Array Mapped Tries

An immutable multi-map is a many-to-many thread-friendly map data structure with expected fast insert and lookup operations. This data structure is used for applications processing graphs or many-to-many relations as applied in static analysis of object-oriented systems. When processing such big data sets the memory overhead of the data structure encoding itself is a memory usage bottleneck. Motivated by reuse and type-safety, libraries for Java, Scala and Clojure typically implement immutable multi-maps by nesting sets as the values with the keys of a trie map. Like this, based on our measurements the expected byte overhead for a sparse multi-map per stored entry adds up to around 65B, which renders it unfeasible to compute with effectively on the JVM. In this paper we propose a general framework for Hash-Array Mapped Tries on the JVM which can store type-heterogeneous keys and values: a Heterogeneous Hash-Array Mapped Trie (HHAMT). Among other applications, this allows for a highly efficient multi-map encoding by (a) not reserving space for empty value sets and (b) inlining the values of singleton sets while maintaining a (c) type-safe API. We detail the necessary encoding and optimizations to mitigate the overhead of storing and retrieving heterogeneous data in a hash-trie. Furthermore, we evaluate HHAMT specifically for the application to multi-maps, comparing them to state-of-the-art encodings of multi-maps in Java, Scala and Clojure. We isolate key differences using microbenchmarks and validate the resulting conclusions on a real world case in static analysis. The new encoding brings the per key-value storage overhead down to 30B: a 2x improvement. With additional inlining of primitive values it reaches a 4x improvement

AN INVESTIGATION OF METAHEURISTICS USING PATH- RELINKING ON THE QUADRATIC ASSIGNMENT PROBLEM

The Quadratic Assignment Problem (QAP) is a widely researched, yet complex, combinatorial optimization problem that is applicable in modeling many real-world problems. Specifically, many optimization problems are formulated as QAPs. To resolve QAPs, the recent trends have been to use metaheuristics rather than exact or heuristic methods, and many researchers have found that the use of hybrid metaheuristics is actually more effective. A newly proposed hybrid metaheuristic is path relinking (PR), which is used to generate solutions by combining two or more reference solutions. In this dissertation, we investigated these diversification and intensification mechanisms using QAP. To satisfy the extensive demands of the computational resources, we utilized a High Throughput Computing (HTC) environment and test cases from the QAPLIB (QAP test case repository). This dissertation consists of three integrated studies that are built upon each other. The first phase explores the effects of the parameter tuning, metaheuristic design, and representation schemes (random keys and permutation solution encoding procedures) of two path-based metaheuristics (Tabu Search and Simulated Annealing) and two population-based metaheuristics (Genetic Algorithms and Artificial Immune Algorithms) using QAP as a testbed. In the second phase of the study, we examined eight tuned metaheuristics representing two representation schemes using problem characteristics. We use problem size, flow and distance dominance measures, sparsity (number of zero entries in the matrices), and the coefficient of correlation measures of the matrices to build search trajectories. The third phase of the dissertation focuses on intensification and diversification mechanisms using path-relinking (PR) procedures (the two variants of position-based path relinking) to enhance the performance of path-based and population-based metaheuristics. The current research in this field has explored the unusual effectiveness of PR algorithms in variety of applications and has emphasized the significance of future research incorporating more sophisticated strategies and frameworks. In addition to addressing these issues, we also examined the effects of solution representations on PR augmentation. For future research, we propose metaheuristic studies using fitness landscape analysis to investigate particular metaheuristics\u27 fitness landscapes and evolution through parameter tuning, solution representation, and PR augmentation. The main research contributions of this dissertation are to widen the knowledge domains of metaheuristic design, representation schemes, parameter tuning, PR mechanism viability, and search trajectory analysis of the fitness landscape using QAPs

Online Mixed-Integer Optimization in Milliseconds

We propose a method to solve online mixed-integer optimization (MIO) problems at very high speed using machine learning. By exploiting the repetitive nature of online optimization, we are able to greatly speedup the solution time. Our approach encodes the optimal solution into a small amount of information denoted as strategy using the Voice of Optimization framework proposed in [BS21]. In this way the core part of the optimization algorithm becomes a multiclass classification problem which can be solved very quickly. In this work, we extend that framework to real-time and high-speed applications focusing on parametric mixed-integer quadratic optimization (MIQO). We propose an extremely fast online optimization algorithm consisting of a feedforward neural network (NN) evaluation and a linear system solution where the matrix has already been factorized. Therefore, this online approach does not require any solver nor iterative algorithm. We show the speed of the proposed method both in terms of total computations required and measured execution time. We estimate the number of floating point operations (flops) required to completely recover the optimal solution as a function of the problem dimensions. Compared to state-of-the-art MIO routines, the online running time of our method is very predictable and can be lower than a single matrix factorization time. We benchmark our method against the state-of-the-art solver Gurobi obtaining from two to three orders of magnitude speedups on examples from fuel cell energy management, sparse portfolio optimization and motion planning with obstacle avoidance

An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure

We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common theory, leading to novel methods for working with several important models of interest in statistics, machine learning and computer vision. In Part 1, we review the concepts of network flows and submodular function optimization theory foundational to our results. We then examine the connections between network flows and the minimum-norm algorithm from submodular optimization, extending and improving several current results. This leads to a concise representation of the structure of a large class of pairwise regularized models important in machine learning, statistics and computer vision. In Part 2, we describe the full regularization path of a class of penalized regression problems with dependent variables that includes the graph-guided LASSO and total variation constrained models. This description also motivates a practical algorithm. This allows us to efficiently find the regularization path of the discretized version of TV penalized models. Ultimately, our new algorithms scale up to high-dimensional problems with millions of variables

Survey on Combinatorial Register Allocation and Instruction Scheduling

Register allocation (mapping variables to processor registers or memory) and instruction scheduling (reordering instructions to increase instruction-level parallelism) are essential tasks for generating efficient assembly code in a compiler. In the last three decades, combinatorial optimization has emerged as an alternative to traditional, heuristic algorithms for these two tasks. Combinatorial optimization approaches can deliver optimal solutions according to a model, can precisely capture trade-offs between conflicting decisions, and are more flexible at the expense of increased compilation time. This paper provides an exhaustive literature review and a classification of combinatorial optimization approaches to register allocation and instruction scheduling, with a focus on the techniques that are most applied in this context: integer programming, constraint programming, partitioned Boolean quadratic programming, and enumeration. Researchers in compilers and combinatorial optimization can benefit from identifying developments, trends, and challenges in the area; compiler practitioners may discern opportunities and grasp the potential benefit of applying combinatorial optimization