Learning the Solution Manifold in Optimization and Its Application in Motion Planning2020-07-24 ${\displaystyle \cong }$ |

Optimization is an essential component for solving problems in wide-ranging fields. Ideally, the objective function should be designed such that the solution is unique and the optimization problem can be solved stably. However, the objective function used in a practical application is usually non-convex, and sometimes it even has an infinite set of solutions. To address this issue, we propose to learn the solution manifold in optimization. We train a model conditioned on the latent variable such that the model represents an infinite set of solutions. In our framework, we reduce this problem to density estimation by using importance sampling, and the latent representation of the solutions is learned by maximizing the variational lower bound. We apply the proposed algorithm to motion-planning problems, which involve the optimization of high-dimensional parameters. The experimental results indicate that the solution manifold can be learned with the proposed algorithm, and the trained model represents an infinite set of homotopic solutions for motion-planning problems. |

Discovering Diverse Solutions in Deep Reinforcement Learning2021-03-11 ${\displaystyle \cong }$ |

Reinforcement learning (RL) algorithms are typically limited to learning a single solution of a specified task, even though there often exists diverse solutions to a given task. Compared with learning a single solution, learning a set of diverse solutions is beneficial because diverse solutions enable robust few-shot adaptation and allow the user to select a preferred solution. Although previous studies have showed that diverse behaviors can be modeled with a policy conditioned on latent variables, an approach for modeling an infinite set of diverse solutions with continuous latent variables has not been investigated. In this study, we propose an RL method that can learn infinitely many solutions by training a policy conditioned on a continuous or discrete low-dimensional latent variable. Through continuous control tasks, we demonstrate that our method can learn diverse solutions in a data-efficient manner and that the solutions can be used for few-shot adaptation to solve unseen tasks. |

Augmenting High-dimensional Nonlinear Optimization with Conditional GANs2021-02-20 ${\displaystyle \cong }$ |

Many mathematical optimization algorithms fail to sufficiently explore the solution space of high-dimensional nonlinear optimization problems due to the curse of dimensionality. This paper proposes generative models as a complement to optimization algorithms to improve performance in problems with high dimensionality. To demonstrate this method, a conditional generative adversarial network (C-GAN) is used to augment the solutions produced by a genetic algorithm (GA) for a 311-dimensional nonconvex multi-objective mixed-integer nonlinear optimization. The C-GAN, composed of two networks with three fully connected hidden layers, is trained on solutions generated by the GA, and then given sets of desired labels (i.e., objective function values), generates complementary solutions corresponding to those labels. Six experiments are conducted to evaluate the capabilities of the proposed method. The generated complementary solutions are compared to the original solutions in terms of optimality and diversity. The generative model generates solutions with objective functions up to 100% better, and with hypervolumes up to 100% higher, than the original solutions. These findings show that a C-GAN with even a simple training approach and simple architecture can highly improve the diversity and optimality of solutions found by an optimization algorithm for a high-dimensional nonlinear optimization problem. |

Optimizing Wireless Systems Using Unsupervised and Reinforced-Unsupervised Deep Learning2020-01-03 ${\displaystyle \cong }$ |

Resource allocation and transceivers in wireless networks are usually designed by solving optimization problems subject to specific constraints, which can be formulated as variable or functional optimization. If the objective and constraint functions of a variable optimization problem can be derived, standard numerical algorithms can be applied for finding the optimal solution, which however incur high computational cost when the dimension of the variable is high. To reduce the on-line computational complexity, learning the optimal solution as a function of the environment's status by deep neural networks (DNNs) is an effective approach. DNNs can be trained under the supervision of optimal solutions, which however, is not applicable to the scenarios without models or for functional optimization where the optimal solutions are hard to obtain. If the objective and constraint functions are unavailable, reinforcement learning can be applied to find the solution of a functional optimization problem, which is however not tailored to optimization problems in wireless networks. In this article, we introduce unsupervised and reinforced-unsupervised learning frameworks for solving both variable and functional optimization problems without the supervision of the optimal solutions. When the mathematical model of the environment is completely known and the distribution of environment's status is known or unknown, we can invoke unsupervised learning algorithm. When the mathematical model of the environment is incomplete, we introduce reinforced-unsupervised learning algorithms that learn the model by interacting with the environment. Our simulation results confirm the applicability of these learning frameworks by taking a user association problem as an example. |

Oracle-Based Robust Optimization via Online Learning2014-02-25 ${\displaystyle \cong }$ |

Robust optimization is a common framework in optimization under uncertainty when the problem parameters are not known, but it is rather known that the parameters belong to some given uncertainty set. In the robust optimization framework the problem solved is a min-max problem where a solution is judged according to its performance on the worst possible realization of the parameters. In many cases, a straightforward solution of the robust optimization problem of a certain type requires solving an optimization problem of a more complicated type, and in some cases even NP-hard. For example, solving a robust conic quadratic program, such as those arising in robust SVM, ellipsoidal uncertainty leads in general to a semidefinite program. In this paper we develop a method for approximately solving a robust optimization problem using tools from online convex optimization, where in every stage a standard (non-robust) optimization program is solved. Our algorithms find an approximate robust solution using a number of calls to an oracle that solves the original (non-robust) problem that is inversely proportional to the square of the target accuracy. |

PAMELI: A Meta-Algorithm for Computationally Expensive Multi-Objective Optimization Problems2021-03-19 ${\displaystyle \cong }$ |

We present an algorithm for multi-objective optimization of computationally expensive problems. The proposed algorithm is based on solving a set of surrogate problems defined by models of the real one, so that only solutions estimated to be approximately Pareto-optimal are evaluated using the real expensive functions. Aside of the search for solutions, our algorithm also performs a meta-search for optimal surrogate models and navigation strategies for the optimization landscape, therefore adapting the search strategy for solutions to the problem as new information about it is obtained. The competitiveness of our approach is demonstrated by an experimental comparison with one state-of-the-art surrogate-assisted evolutionary algorithm on a set of benchmark problems. |

Instance Specific Approximations for Submodular Maximization2021-02-23 ${\displaystyle \cong }$ |

For many optimization problems in machine learning, finding an optimal solution is computationally intractable and we seek algorithms that perform well in practice. Since computational intractability often results from pathological instances, we look for methods to benchmark the performance of algorithms against optimal solutions on real-world instances. The main challenge is that an optimal solution cannot be efficiently computed for intractable problems, and we therefore often do not know how far a solution is from being optimal. A major question is therefore how to measure the performance of an algorithm in comparison to an optimal solution on instances we encounter in practice. In this paper, we address this question in the context of submodular optimization problems. For the canonical problem of submodular maximization under a cardinality constraint, it is intractable to compute a solution that is better than a $1-1/e \approx 0.63$ fraction of the optimum. Algorithms like the celebrated greedy algorithm are guaranteed to achieve this $1-1/e$ bound on any instance and are used in practice. Our main contribution is not a new algorithm for submodular maximization but an analytical method that measures how close an algorithm for submodular maximization is to optimal on a given problem instance. We use this method to show that on a wide variety of real-world datasets and objectives, the approximation of the solution found by greedy goes well beyond $1-1/e$ and is often at least 0.95. We develop this method using a novel technique that lower bounds the objective of a dual minimization problem to obtain an upper bound on the value of an optimal solution to the primal maximization problem. |

Reinforcement Learning for Solving the Vehicle Routing Problem2018-05-21 ${\displaystyle \cong }$ |

We present an end-to-end framework for solving the Vehicle Routing Problem (VRP) using reinforcement learning. In this approach, we train a single model that finds near-optimal solutions for problem instances sampled from a given distribution, only by observing the reward signals and following feasibility rules. Our model represents a parameterized stochastic policy, and by applying a policy gradient algorithm to optimize its parameters, the trained model produces the solution as a sequence of consecutive actions in real time, without the need to re-train for every new problem instance. On capacitated VRP, our approach outperforms classical heuristics and Google's OR-Tools on medium-sized instances in solution quality with comparable computation time (after training). We demonstrate how our approach can handle problems with split delivery and explore the effect of such deliveries on the solution quality. Our proposed framework can be applied to other variants of the VRP such as the stochastic VRP, and has the potential to be applied more generally to combinatorial optimization problems. |

Differentially Private Convex Optimization with Feasibility Guarantees2020-06-22 ${\displaystyle \cong }$ |

This paper develops a novel differentially private framework to solve convex optimization problems with sensitive optimization data and complex physical or operational constraints. Unlike standard noise-additive algorithms, that act primarily on the problem data, objective or solution, and disregard the problem constraints, this framework requires the optimization variables to be a function of the noise and exploits a chance-constrained problem reformulation with formal feasibility guarantees. The noise is calibrated to provide differential privacy for identity and linear queries on the optimization solution. For many applications, including resource allocation problems, the proposed framework provides a trade-off between the expected optimality loss and the variance of optimization results. |

Online Mixed-Integer Optimization in Milliseconds2020-06-29 ${\displaystyle \cong }$ |

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 [BS20]. 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. |

Predicting Tactical Solutions to Operational Planning Problems under Imperfect Information2020-03-19 ${\displaystyle \cong }$ |

This paper offers a methodological contribution at the intersection of machine learning and operations research. Namely, we propose a methodology to quickly predict tactical solutions to a given operational problem. In this context, the tactical solution is less detailed than the operational one but it has to be computed in very short time and under imperfect information. The problem is of importance in various applications where tactical and operational planning problems are interrelated and information about the operational problem is revealed over time. This is for instance the case in certain capacity planning and demand management systems. We formulate the problem as a two-stage optimal prediction stochastic program whose solution we predict with a supervised machine learning algorithm. The training data set consists of a large number of deterministic (second stage) problems generated by controlled probabilistic sampling. The labels are computed based on solutions to the deterministic problems (solved independently and offline) employing appropriate aggregation and subselection methods to address uncertainty. Results on our motivating application in load planning for rail transportation show that deep learning algorithms produce highly accurate predictions in very short computing time (milliseconds or less). The prediction accuracy is comparable to solutions computed by sample average approximation of the stochastic program. |

A Novel Meta-Heuristic Optimization Algorithm Inspired by the Spread of Viruses2020-06-11 ${\displaystyle \cong }$ |

According to the no-free-lunch theorem, there is no single meta-heuristic algorithm that can optimally solve all optimization problems. This motivates many researchers to continuously develop new optimization algorithms. In this paper, a novel nature-inspired meta-heuristic optimization algorithm called virus spread optimization (VSO) is proposed. VSO loosely mimics the spread of viruses among hosts, and can be effectively applied to solving many challenging and continuous optimization problems. We devise a new representation scheme and viral operations that are radically different from previously proposed virus-based optimization algorithms. First, the viral RNA of each host in VSO denotes a potential solution for which different viral operations will help to diversify the searching strategies in order to largely enhance the solution quality. In addition, an imported infection mechanism, inheriting the searched optima from another colony, is introduced to possibly avoid the prematuration of any potential solution in solving complex problems. VSO has an excellent capability to conduct adaptive neighborhood searches around the discovered optima for achieving better solutions. Furthermore, with a flexible infection mechanism, VSO can quickly escape from local optima. To clearly demonstrate both its effectiveness and efficiency, VSO is critically evaluated on a series of well-known benchmark functions. Moreover, VSO is validated on its applicability through two real-world examples including the financial portfolio optimization and optimization of hyper-parameters of support vector machines for classification problems. The results show that VSO has attained superior performance in terms of solution fitness, convergence rate, scalability, reliability, and flexibility when compared to those results of the conventional as well as state-of-the-art meta-heuristic optimization algorithms. |

Learning to Optimize with Unsupervised Learning: Training Deep Neural Networks for URLLC2019-05-27 ${\displaystyle \cong }$ |

Learning the optimized solution as a function of environmental parameters is effective in solving numerical optimization in real time for time-sensitive applications. Existing works of learning to optimize train deep neural networks (DNN) with labels, and the learnt solution are inaccurate, which cannot be employed to ensure the stringent quality of service. In this paper, we propose a framework to learn the latent function with unsupervised deep learning, where the property that the optimal solution should satisfy is used as the "supervision signal" implicitly. The framework is applicable to both functional and variable optimization problems with constraints. We take a variable optimization problem in ultra-reliable and low-latency communications as an example, which demonstrates that the ultra-high reliability can be supported by the DNN without supervision labels. |

A Coordinate-wise Optimization Algorithm for Sparse Inverse Covariance Selection2018-04-04 ${\displaystyle \cong }$ |

Sparse inverse covariance selection is a fundamental problem for analyzing dependencies in high dimensional data. However, such a problem is difficult to solve since it is NP-hard. Existing solutions are primarily based on convex approximation and iterative hard thresholding, which only lead to sub-optimal solutions. In this work, we propose a coordinate-wise optimization algorithm to solve this problem which is guaranteed to converge to a coordinate-wise minimum point. The algorithm iteratively and greedily selects one variable or swaps two variables to identify the support set, and then solves a reduced convex optimization problem over the support set to achieve the greatest descent. As a side contribution of this paper, we propose a Newton-like algorithm to solve the reduced convex sub-problem, which is proven to always converge to the optimal solution with global linear convergence rate and local quadratic convergence rate. Finally, we demonstrate the efficacy of our method on synthetic data and real-world data sets. As a result, the proposed method consistently outperforms existing solutions in terms of accuracy. |

Principal Component Analysis Applied to Gradient Fields in Band Gap Optimization Problems for Metamaterials2021-04-04 ${\displaystyle \cong }$ |

A promising technique for the spectral design of acoustic metamaterials is based on the formulation of suitable constrained nonlinear optimization problems. Unfortunately, the straightforward application of classical gradient-based iterative optimization algorithms to the numerical solution of such problems is typically highly demanding, due to the complexity of the underlying physical models. Nevertheless, supervised machine learning techniques can reduce such a computational effort, e.g., by replacing the original objective functions of such optimization problems with more-easily computable approximations. In this framework, the present article describes the application of a related unsupervised machine learning technique, namely, principal component analysis, to approximate the gradient of the objective function of a band gap optimization problem for an acoustic metamaterial, with the aim of making the successive application of a gradient-based iterative optimization algorithm faster. Numerical results show the effectiveness of the proposed method. |

A PAC algorithm in relative precision for bandit problem with costly sampling2020-07-30 ${\displaystyle \cong }$ |

This paper considers the problem of maximizing an expectation function over a finite set, or finite-arm bandit problem. We first propose a naive stochastic bandit algorithm for obtaining a probably approximately correct (PAC) solution to this discrete optimization problem in relative precision, that is a solution which solves the optimization problem up to a relative error smaller than a prescribed tolerance, with high probability. We also propose an adaptive stochastic bandit algorithm which provides a PAC-solution with the same guarantees. The adaptive algorithm outperforms the mean complexity of the naive algorithm in terms of number of generated samples and is particularly well suited for applications with high sampling cost. |

Learning to Optimize Under Constraints with Unsupervised Deep Neural Networks2021-01-03 ${\displaystyle \cong }$ |

In this paper, we propose a machine learning (ML) method to learn how to solve a generic constrained continuous optimization problem. To the best of our knowledge, the generic methods that learn to optimize, focus on unconstrained optimization problems and those dealing with constrained problems are not easy-to-generalize. This approach is quite useful in optimization tasks where the problem's parameters constantly change and require resolving the optimization task per parameter update. In such problems, the computational complexity of optimization algorithms such as gradient descent or interior point method preclude near-optimal designs in real-time applications. In this paper, we propose an unsupervised deep learning (DL) solution for solving constrained optimization problems in real-time by relegating the main computation load to offline training phase. This paper's main contribution is proposing a method for enforcing the equality and inequality constraints to the DL-generated solutions for generic optimization tasks. |

Warm Starting Bayesian Optimization2016-08-11 ${\displaystyle \cong }$ |

We develop a framework for warm-starting Bayesian optimization, that reduces the solution time required to solve an optimization problem that is one in a sequence of related problems. This is useful when optimizing the output of a stochastic simulator that fails to provide derivative information, for which Bayesian optimization methods are well-suited. Solving sequences of related optimization problems arises when making several business decisions using one optimization model and input data collected over different time periods or markets. While many gradient-based methods can be warm started by initiating optimization at the solution to the previous problem, this warm start approach does not apply to Bayesian optimization methods, which carry a full metamodel of the objective function from iteration to iteration. Our approach builds a joint statistical model of the entire collection of related objective functions, and uses a value of information calculation to recommend points to evaluate. |

Automatically Learning Compact Quality-aware Surrogates for Optimization Problems2020-06-18 ${\displaystyle \cong }$ |

Solving optimization problems with unknown parameters often requires learning a predictive model to predict the values of the unknown parameters and then solving the problem using these values. Recent work has shown that including the optimization problem as a layer in the model training pipeline results in predictions of the unobserved parameters that lead to higher decision quality. Unfortunately, this process comes at a large computational cost because the optimization problem must be solved and differentiated through in each training iteration; furthermore, it may also sometimes fail to improve solution quality due to non-smoothness issues that arise when training through a complex optimization layer. To address these shortcomings, we learn a low-dimensional surrogate model of a large optimization problem by representing the feasible space in terms of meta-variables, each of which is a linear combination of the original variables. By training a low-dimensional surrogate model end-to-end, and jointly with the predictive model, we achieve: i) a large reduction in training and inference time; and ii) improved performance by focusing attention on the more important variables in the optimization and learning in a smoother space. Empirically, we demonstrate these improvements on a non-convex adversary modeling task, a submodular recommendation task and a convex portfolio optimization task. |

Sparse Approximate Solutions to Max-Plus Equations with Application to Multivariate Convex Regression2020-11-06 ${\displaystyle \cong }$ |

In this work, we study the problem of finding approximate, with minimum support set, solutions to matrix max-plus equations, which we call sparse approximate solutions. We show how one can obtain such solutions efficiently and in polynomial time for any $\ell_p$ approximation error. Based on these results, we propose a novel method for piecewise-linear fitting of convex multivariate functions, with optimality guarantees for the model parameters and an approximately minimum number of affine regions. |