We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2=Y$, the matrix analog of binary variables that satisfy $z^2=z$, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields tractable convex optimization problems over the non-convex set of orthogonal projection matrices. Furthermore, we design outer-approximation algorithms to solve low-rank problems to certifiable optimality, compute lower bounds via their semidefinite relaxations, and provide near optimal solutions through rounding and local search techniques. We implement these numerical ingredients and, for the first time, solve low-rank optimization problems to certifiable optimality. Our algorithms also supply certifiably near-optimal solutions for larger problem sizes and outperform existing heuristics, by deriving an alternative to the popular nuclear norm relaxation which generalizes the perspective relaxation from vectors to matrices. All in all, our framework, which we name Mixed-Projection Conic Optimization, solves low-rank problems to certifiable optimality in a tractable and unified fashion.

The network embedding problem that maps nodes in a graph to vectors in Euclidean space can be very useful for addressing several important tasks on a graph. Recently, graph neural networks (GNNs) have been proposed for solving such a problem. However, most embedding algorithms and GNNs are difficult to interpret and do not scale well to handle millions of nodes. In this paper, we tackle the problem from a new perspective based on the equivalence of three constrained optimization problems: the network embedding problem, the trace maximization problem of the modularity matrix in a sampled graph, and the matrix factorization problem of the modularity matrix in a sampled graph. The optimal solutions to these three problems are the dominant eigenvectors of the modularity matrix. We proposed two algorithms that belong to a special class of graph convolutional networks (GCNs) for solving these problems: (i) Clustering As Feature Embedding GCN (CAFE-GCN) and (ii) sphere-GCN. Both algorithms are stable trace maximization algorithms, and they yield good approximations of dominant eigenvectors. Moreover, there are linear-time implementations for sparse graphs. In addition to solving the network embedding problem, both proposed GCNs are capable of performing dimensionality reduction. Various experiments are conducted to evaluate our proposed GCNs and show that our proposed GCNs outperform almost all the baseline methods. Moreover, CAFE-GCN could be benefited from the labeled data and have tremendous improvements in various performance metrics.

We propose a novel method for exploring the dynamics of physically based animated characters, and learning a task-agnostic action space that makes movement optimization easier. Like several previous papers, we parameterize actions as target states, and learn a short-horizon goal-conditioned low-level control policy that drives the agent's state towards the targets. Our novel contribution is that with our exploration data, we are able to learn the low-level policy in a generic manner and without any reference movement data. Trained once for each agent or simulation environment, the policy improves the efficiency of optimizing both trajectories and high-level policies across multiple tasks and optimization algorithms. We also contribute novel visualizations that show how using target states as actions makes optimized trajectories more robust to disturbances; this manifests as wider optima that are easy to find. Due to its simplicity and generality, our proposed approach should provide a building block that can improve a large variety of movement optimization methods and applications.

This work formulates antiviral repositioning as a matrix completion problem where the antiviral drugs are along the rows and the viruses along the columns. The input matrix is partially filled, with ones in positions where the antiviral has been known to be effective against a virus. The curated metadata for antivirals (chemical structure and pathways) and viruses (genomic structure and symptoms) is encoded into our matrix completion framework as graph Laplacian regularization. We then frame the resulting multiple graph regularized matrix completion problem as deep matrix factorization. This is solved by using a novel optimization method called HyPALM (Hybrid Proximal Alternating Linearized Minimization). Results on our curated RNA drug virus association (DVA) dataset shows that the proposed approach excels over state-of-the-art graph regularized matrix completion techniques. When applied to "in silico" prediction of antivirals for COVID-19, our approach returns antivirals that are either used for treating patients or are under for trials for the same.

We focus on the problem of finding an optimal strategy for a team of two players that faces an opponent in an imperfect-information zero-sum extensive-form game. Team members are not allowed to communicate during play but can coordinate before the game. In that setting, it is known that the best the team can do is sample a profile of potentially randomized strategies (one per player) from a joint (a.k.a. correlated) probability distribution at the beginning of the game. In this paper, we first provide new modeling results about computing such an optimal distribution by drawing a connection to a different literature on extensive-form correlation. Second, we provide an algorithm that computes such an optimal distribution by only using profiles where only one of the team members gets to randomize in each profile. We can also cap the number of such profiles we allow in the solution. This begets an anytime algorithm by increasing the cap. We find that often a handful of well-chosen such profiles suffices to reach optimal utility for the team. This enables team members to reach coordination through a relatively simple and understandable plan. Finally, inspired by this observation and leveraging theoretical concepts that we introduce, we develop an efficient column-generation algorithm for finding an optimal distribution for the team. We evaluate it on a suite of common benchmark games. It is three orders of magnitude faster than the prior state of the art on games that the latter can solve and it can also solve several games that were previously unsolvable.

We propose a generative Causal Adversarial Network (CAN) for learning and sampling from conditional and interventional distributions. In contrast to the existing CausalGAN which requires the causal graph to be given, our proposed framework learns the causal relations from the data and generates samples accordingly. The proposed CAN comprises a two-fold process namely Label Generation Network (LGN) and Conditional Image Generation Network (CIGN). The LGN is a GAN-based architecture which learns and samples from the causal model over labels. The sampled labels are then fed to CIGN, a conditional GAN architecture, which learns the relationships amongst labels and pixels and pixels themselves and generates samples based on them. This framework is equipped with an intervention mechanism which enables. the model to generate samples from interventional distributions. We quantitatively and qualitatively assess the performance of CAN and empirically show that our model is able to generate both interventional and conditional samples without having access to the causal graph for the application of face generation on CelebA data.

Initial development and subsequent calibration of discrete event simulation models for complex systems require accurate identification of dynamically changing process characteristics. Existing data driven change point methods (DD-CPD) assume changes are extraneous to the system, thus cannot utilize available process knowledge. This work proposes a unified framework for process-driven multivariate change point detection (PD-CPD) by combining change point detection models with machine learning and process-driven simulation modeling. The PD-CPD, after initializing with DD-CPD's change point(s), uses simulation models to generate system level outputs as time-series data streams which are then used to train neural network models to predict system characteristics and change points. The accuracy of the predictive models measures the likelihood that the actual process data conforms to the simulated change points in system characteristics. PD-CPD iteratively optimizes change points by repeating simulation and predictive model building steps until the set of change point(s) with the maximum likelihood is identified. Using an emergency department case study, we show that PD-CPD significantly improves change point detection accuracy over DD-CPD estimates and is able to detect actual change points. This work has been submitted to the IEEE for possible publication. Increasing complexity of modern systems require a new generation of simulation models that can accurately represent the time-varying system dynamics and provide decision support. In manufacturing and service systems, discrete event simulation (DES) models are extensively used to represent the discrete flow of materials, requests and customers in dynamic environments. Challenges associated with the complex nature of modern systems are being increasingly addressed by digital twin technologies [1], [2] that aim to create a one-to-one replica of the physical world using highly detailed simulation models.

Optimization Modeling in Python, Pyomo models with Jupyter Notebooks Created by A. Soroudi Preview this Course GET COUPON CODE **Brand New For September 2020 - Optimization modeling in Python Course on Udemy** Join your fellow researchers and experts in operation research industry in learning the fundamentals of the optimal decision making and optimization . I will walk you through every step of Python coding with real-life case studies, actual experiments, and tons of examples from around different disciplines. By the end of this course, you'll be able to: Code your own optimization problem in Python. Receive your official certificate The developed course is suitable for you even if you have no background in the power systems. In this Optimization in Python from scratch course you will learn: How to formulate your problem and implement it in Python and make optimal decisions in your real-life problems How to code efficiently, get familiarised with the techniques that will make your code scalable for large problems How to design an action block with a clearly defined conversion goal How to run sensitivity analysis in Python to predict the outcome of a decision if a situation turns out to be different compared to the key predictions.

Black-box artificial intelligence (AI) induction methods such as deep reinforcement learning (DRL) are increasingly being used to find optimal policies for a given control task. Although policies represented using a black-box AI are capable of efficiently executing the underlying control task and achieving optimal closed-loop performance -- controlling the agent from initial time step until the successful termination of an episode, the developed control rules are often complex and neither interpretable nor explainable. In this paper, we use a recently proposed nonlinear decision-tree (NLDT) approach to find a hierarchical set of control rules in an attempt to maximize the open-loop performance for approximating and explaining the pre-trained black-box DRL (oracle) agent using the labelled state-action dataset. Recent advances in nonlinear optimization approaches using evolutionary computation facilitates finding a hierarchical set of nonlinear control rules as a function of state variables using a computationally fast bilevel optimization procedure at each node of the proposed NLDT. Additionally, we propose a re-optimization procedure for enhancing closed-loop performance of an already derived NLDT. We evaluate our proposed methodologies on four different control problems having two to four discrete actions. In all these problems our proposed approach is able to find simple and interpretable rules involving one to four non-linear terms per rule, while simultaneously achieving on par closed-loop performance when compared to a trained black-box DRL agent. The obtained results are inspiring as they suggest the replacement of complicated black-box DRL policies involving thousands of parameters (making them non-interpretable) with simple interpretable policies. Results are encouraging and motivating to pursue further applications of proposed approach in solving more complex control tasks.

When we are faced with a combinatorial optimization problem, our goal is to design an algorithm that satisfies three important properties. However, when we deal with an NP-hard problem, it is not at all clear whether such algorithms that satisfy all three aforementioned properties even exist. So, one natural approach to slightly simplify our lives is to relax one of the three properties. In the case of approximation algorithms, we decide to relax property (2). This means that we are interested in efficient algorithms that always return feasible solutions, and what we care to bound is the gap between the value of the solution they return and the value of an optimal solution.