Quality diversity (QD) is a growing branch of stochastic optimization research that studies the problem of generating an archive of solutions that maximize a given objective function but are also diverse with respect to a set of specified measure functions. However, even when these functions are differentiable, QD algorithms treat them as "black boxes", ignoring gradient information. We present the differentiable quality diversity (DQD) problem, a special case of QD, where both the objective and measure functions are first order differentiable. We then present MAP-Elites via Gradient Arborescence (MEGA), a DQD algorithm that leverages gradient information to efficiently explore the joint range of the objective and measure functions. Results in two QD benchmark domains and in searching the latent space of a StyleGAN show that MEGA significantly outperforms state-of-the-art QD algorithms, highlighting DQD's promise for efficient quality diversity optimization when gradient information is available. Source code is available at https://github.com/icaros-usc/dqd.

Inverse molecular design, i.e., designing molecules with specific target properties, can be posed as an optimization problem. High-dimensional optimization tasks in the natural sciences are commonly tackled via population-based metaheuristic optimization algorithms such as evolutionary algorithms. However, expensive property evaluation, which is often required, can limit the widespread use of such approaches as the associated cost can become prohibitive. Herein, we present JANUS, a genetic algorithm that is inspired by parallel tempering. It propagates two populations, one for exploration and another for exploitation, improving optimization by reducing expensive property evaluations. Additionally, JANUS is augmented by a deep neural network that approximates molecular properties via active learning for enhanced sampling of the chemical space. Our method uses the SELFIES molecular representation and the STONED algorithm for the efficient generation of structures, and outperforms other generative models in common inverse molecular design tasks achieving state-of-the-art performance.

How to become pro in Linear Programming for Data Science? In this course you will learn all about the mathematical optimization of linear programming in data science. This course is very unique and have its own importance in their respective disciplines. The data science and business study heavily rely on optimization. Optimization is the study of analysis and interpreting mathematical data under the special rules and formula.

Machine learning involves using an algorithm to learn and generalize from historical data in order to make predictions on new data. This problem can be described as approximating a function that maps examples of inputs to examples of outputs. Approximating a function can be solved by framing the problem as function optimization. This is where a machine learning algorithm defines a parameterized mapping function (e.g. a weighted sum of inputs) and an optimization algorithm is used to fund the values of the parameters (e.g. This means that each time we fit a machine learning algorithm on a training dataset, we are solving an optimization problem.

In practical applications, there can be significantly more than three objectives accounting for a mix of performance, resource, environmental and other goals; for example, Kollat et al. (2011) optimize the position and frequency of tracer sampling of groundwater using six design objectives, Eikelboom et al. (2015) address land use planning problems using cost, benefit and spatial objectives for the individual parcels of land, and Fleming et al. (2005) design optimal control systems for aircraft engines using eight objectives, each corresponding to a different design specifications. Multi-objective optimization (MO) (Deb, 2001; Miettinen, 2012) is the area looking at the development and application of algorithms to problems with multiple conflicting objectives; problems with more than three objectives have also been termed as many-objective problems (Kollat et al., 2011) and are less studied. In the absence of any user preferences about desired ideal solutions, the goal of MO algorithms (MOAs) is not to identify a single optimal solution but to approximate the set of best trade-off solutions to a problem, also known as the Pareto (optimal) set. We say that a solution Pareto dominates, or simply dominates, another solution if it is not worse in any objective and if it is strictly better in at least one objective. Solutions in the Pareto set are non-dominated by any other solution from the feasible solution space. The traditional approach to tackle a MO problem is to convert it into a single-objective problem using a scalarizing function (Eichfelder, 2008), and then solve the problem repeatedly using different'configurations' of the scalarizing function (e.g.

PYROBOCOP is a lightweight Python-based package for control and optimization of robotic systems described by nonlinear Differential Algebraic Equations (DAEs). In particular, the package can handle systems with contacts that are described by complementarity constraints and provides a general framework for specifying obstacle avoidance constraints. The package performs direct transcription of the DAEs into a set of nonlinear equations by performing orthogonal collocation on finite elements. The resulting optimization problem belongs to the class of Mathematical Programs with Complementarity Constraints (MPCCs). MPCCs fail to satisfy commonly assumed constraint qualifications and require special handling of the complementarity constraints in order for NonLinear Program (NLP) solvers to solve them effectively. PYROBOCOP provides automatic reformulation of the complementarity constraints that enables NLP solvers to perform optimization of robotic systems. The package is interfaced with ADOLC for obtaining sparse derivatives by automatic differentiation and IPOPT for performing optimization. We demonstrate the effectiveness of our approach in terms of speed and flexibility. We provide several numerical examples for several robotic systems with collision avoidance as well as contact constraints represented using complementarity constraints. We provide comparisons with other open source optimization packages like CasADi and Pyomo .

Black-box optimization (BBO) has a broad range of applications, including automatic machine learning, engineering, physics, and experimental design. However, it remains a challenge for users to apply BBO methods to their problems at hand with existing software packages, in terms of applicability, performance, and efficiency. In this paper, we build OpenBox, an open-source and general-purpose BBO service with improved usability. The modular design behind OpenBox also facilitates flexible abstraction and optimization of basic BBO components that are common in other existing systems. OpenBox is distributed, fault-tolerant, and scalable. To improve efficiency, OpenBox further utilizes "algorithm agnostic" parallelization and transfer learning. Our experimental results demonstrate the effectiveness and efficiency of OpenBox compared to existing systems.

In forming learning objectives, one oftentimes needs to aggregate a set of individual values to a single output. Such cases occur in the aggregate loss, which combines individual losses of a learning model over each training sample, and in the individual loss for multi-label learning, which combines prediction scores over all class labels. In this work, we introduce the sum of ranked range (SoRR) as a general approach to form learning objectives. A ranked range is a consecutive sequence of sorted values of a set of real numbers. The minimization of SoRR is solved with the difference of convex algorithm (DCA). We explore two applications in machine learning of the minimization of the SoRR framework, namely the AoRR aggregate loss for binary/multi-class classification at the sample level and the TKML individual loss for multi-label/multi-class classification at the label level. A combination loss of AoRR and TKML is proposed as a new learning objective for improving the robustness of multi-label learning in the face of outliers in sample and labels alike. Our empirical results highlight the effectiveness of the proposed optimization frameworks and demonstrate the applicability of proposed losses using synthetic and real data sets.

We present gPC-SCP: Generalized Polynomial Chaos-based Sequential Convex Programming method to compute a sub-optimal solution for a continuous-time chance-constrained stochastic nonlinear optimal control problem (SNOC) problem. The approach enables motion planning and control of robotic systems under uncertainty. The proposed method involves two steps. The first step is to derive a deterministic nonlinear optimal control problem (DNOC) with convex constraints that are surrogate to the SNOC by using gPC expansion and the distributionally-robust convex subset of the chance constraints. The second step is to solve the DNOC problem using sequential convex programming (SCP) for trajectory generation and control. We prove that in the unconstrained case, the optimal value of the DNOC converges to that of SNOC asymptotically and that any feasible solution of the constrained DNOC is a feasible solution of the chance-constrained SNOC. We derive a stable stochastic model predictive controller using the gPC-SCP for tracking a trajectory in the presence of uncertainty. We empirically demonstrate the efficacy of the gPC-SCP method for the following three test cases: 1) collision checking under uncertainty in actuation, 2) collision checking with stochastic obstacle model, and 3) safe trajectory tracking under uncertainty in the dynamics and obstacle location by using a receding horizon control approach. We validate the effectiveness of the gPC-SCP method on the robotic spacecraft testbed.

We present an end-to-end framework for the Assignment Problem with multiple tasks mapped to a group of workers, using reinforcement learning while preserving many constraints. Tasks and workers have time constraints and there is a cost associated with assigning a worker to a task. Each worker can perform multiple tasks until it exhausts its allowed time units (capacity). We train a reinforcement learning agent to find near optimal solutions to the problem by minimizing total cost associated with the assignments while maintaining hard constraints. We use proximal policy optimization to optimize model parameters. The model generates a sequence of actions in real-time which correspond to task assignment to workers, without having to retrain for changes in the dynamic state of the environment. In our problem setting reward is computed as negative of the assignment cost. We also demonstrate our results on bin packing and capacitated vehicle routing problem, using the same framework. Our results outperform Google OR-Tools using MIP and CP-SAT solvers with large problem instances, in terms of solution quality and computation time.