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We perform the first tight convergence analysis of the gradient method with varying step sizes when applied to smooth hypoconvex (weakly convex) functions. Hypoconvex functions are smooth nonconvex functions whose curvature is bounded and assumed to belong to the interval $[\mu, L]$, with $\mu<0$. Our convergence rates improve and extend the existing analysis for smooth nonconvex functions with $L$-Lipschitz gradient (which corresponds to the case $\mu=-L$), and smoothly interpolates between that class and the class of smooth convex functions. We obtain our results using the performance estimation framework adapted to hypoconvex functions, for which new interpolation conditions are derived. We derive explicit upper bounds on the minimum gradient norm of the iterates for a large range of step sizes, explain why all such rates share a common structure, and prove that these rates are tight when step sizes are smaller or equal to $1/L$. Finally, we identify the optimal constant step size that minimizes the worst-case of the gradient method applied to hypoconvex functions.

Relative smoothness and strong convexity have recently gained considerable attention in optimization. These notions are generalizations of the classical Euclidean notions of smoothness and strong convexity that are known to be dual to each other. However, conjugate dualities for non-Euclidean relative smoothness and strong convexity remain an open problem as noted earlier by Lu, Freund and Nesterov [SIAM J. Optim., 28 (2018), pp. 333-354]. In this paper we address this question by introducing the notions of anisotropic strong convexity and smoothness as the respective dual counterparts. The dualities are developed under the light of generalized conjugacy which leads us embed the anticipated dual notions within the superclasses of certain upper and lower envelopes. In contrast to the Euclidean case these inclusions are proper in general as showcased by means of counterexamples.

This paper presents alpaqa, an open-source C++ implementation of an augmented Lagrangian method for nonconvex constrained numerical optimization, using the first-order PANOC algorithm as inner solver. The implementation is packaged as an easy-to-use library that can be used in C++ and Python. Furthermore, two improvements to the PANOC algorithm are proposed and their effectiveness is demonstrated in NMPC applications and on the CUTEst benchmarks for numerical optimization. The source code of the alpaqa library is available at https://github.com/kul-optec/alpaqa and binary packages can be installed from https://pypi.org/project/alpaqa ., Comment: Submitted to the 20th European Control Conference (ECC22), London

We consider the problem of interaction-aware motion planning for automated vehicles in general traffic situations. We model the interaction between the controlled vehicle and surrounding road users using a generalized potential game, in which each road user is assumed to minimize a common cost function subject to shared (collision avoidance) constraints. We propose a quadratic penalty method to deal with the shared constraints and solve the resulting optimal control problem online using an Augmented Lagrangian method based on PANOC. Secondly, we present a simple methodology for learning preferences and constraints of other road users online, based on observed behavior. Through extensive simulations in a highway merging scenario, we demonstrate the practical efficacy of the overall approach as well as the benefits of the proposed online learning scheme., Comment: Accepted at 20th European Control Conference (ECC22); extended version

In this paper, we consider a class of nonsmooth nonconvex optimization problems whose objective is the sum of a block relative smooth function and a proper and lower semicontinuous block separable function. Although the analysis of block proximal gradient (BPG) methods for the class of block $L$-smooth functions have been successfully extended to Bregman BPG methods that deal with the class of block relative smooth functions, accelerated Bregman BPG methods are scarce and challenging to design. Taking our inspiration from Nesterov-type acceleration and the majorization-minimization scheme, we propose a block alternating Bregman Majorization-Minimization framework with Extrapolation (BMME). We prove subsequential convergence of BMME to a first-order stationary point under mild assumptions, and study its global convergence under stronger conditions. We illustrate the effectiveness of BMME on the penalized orthogonal nonnegative matrix factorization problem.

Scenario-based stochastic optimal control problems suffer from the curse of dimensionality as they can easily grow to six and seven figure sizes. First-order methods are suitable as they can deal with such large-scale problems, but may fail to achieve accurate solutions within a reasonable number of iterations. To achieve solutions of higher accuracy and high speed, in this paper we propose two proximal quasi-Newtonian limited-memory algorithms - MinFBE applied to the dual problem and the Newton-type alternating minimization algorithm (NAMA) - which can be massively parallelized on lockstep hardware such as graphics processing units (GPUs). We demonstrate the performance of these methods, in terms of convergence speed and parallelizability, on large-scale problems involving millions of variables.

We present a learning model predictive control (MPC) scheme for chance-constrained Markov jump systems with unknown switching probabilities. Using samples of the underlying Markov chain, ambiguity sets of transition probabilities are estimated which include the true conditional probability distributions with high probability. These sets are updated online and used to formulate a time-varying, risk-averse optimal control problem. We prove recursive feasibility of the resulting MPC scheme and show that the original chance constraints remain satisfied at every time step. Furthermore, we show that under sufficient decrease of the confidence levels, the resulting MPC scheme renders the closed-loop system mean-square stable with respect to the true-but-unknown distributions, while remaining less conservative than a fully robust approach. Finally, we show that the value function of the learning MPC converges from above to its nominal counterpart as the sample size grows to infinity. We illustrate our approach on a numerical example., Comment: Revised version

We study safe, data-driven control of (Markov) jump linear systems with unknown transition probabilities, where both the discrete mode and the continuous state are to be inferred from output measurements. To this end, we develop a receding horizon estimator which uniquely identifies a sub-sequence of past mode transitions and the corresponding continuous state, allowing for arbitrary switching behavior. Unlike traditional approaches to mode estimation, we do not require an offline exhaustive search over mode sequences to determine the size of the observation window, but rather select it online. If the system is weakly mode observable, the window size will be upper bounded, leading to a finite-memory observer. We integrate the estimation procedure with a simple distributionally robust controller, which hedges against misestimations of the transition probabilities due to finite sample sizes. As additional mode transitions are observed, the used ambiguity sets are updated, resulting in continual improvements of the control performance. The practical applicability of the approach is illustrated on small numerical examples.

Training of neural networks amounts to nonconvex optimization problems that are typically solved by using backpropagation and (variants of) stochastic gradient descent. In this work we propose an alternative approach by viewing the training task as a nonlinear optimal control problem. Under this lens, backpropagation amounts to the sequential approach (single shooting) to optimal control, where the states variables have been eliminated. It is well known that single shooting may lead to ill conditioning, and for this reason the simultaneous approach (multiple shooting) is typically preferred. Motivated by this hypothesis, an augmented Lagrangian algorithm is developed that only requires an approximate solution to the Lagrangian subproblems up to a user-defined accuracy. By applying this framework to the training of neural networks, it is shown that the inner Lagrangian subproblems are amenable to be solved using Gauss-Newton iterations. To fully exploit the structure of neural networks, the resulting linear least squares problems are addressed by employing an approach based on forward dynamic programming. Finally, the effectiveness of our method is showcased on regression datasets., Comment: 8 pages; typos corrected

In large-scale optimization, the presence of nonsmooth and nonconvex terms in a given problem typically makes it hard to solve. A popular approach to address nonsmooth terms in convex optimization is to approximate them with their respective Moreau envelopes. In this work, we study the use of Lasry-Lions double envelopes to approximate nonsmooth terms that are also not convex. These envelopes are an extension of the Moreau ones but exhibit an additional smoothness property that makes them amenable to fast optimization algorithms. Lasry-Lions envelopes can also be seen as an "intermediate" between a given function and its convex envelope, and we make use of this property to develop a method that builds a sequence of approximate subproblems that are easier to solve than the original problem. We discuss convergence properties of this method when used to address composite minimization problems; additionally, based on a number of experiments, we discuss settings where it may be more useful than classical alternatives in two domains: signal decoding and spectral unmixing., Comment: 29th Eur. Signal Process. Conf. (EUSIPCO 2021), accepted. 5 pages, 2 figures, 2 tables

In this paper we introduce a novel approach to distributionally robust optimal control that supports online learning of the ambiguity set, while guaranteeing recursive feasibility. We introduce conic representable risk, which is useful to derive tractable reformulations of distributionally robust optimization problems. Specifically, to illustrate the techniques introduced, we utilize risk measures constructed based on data-driven ambiguity sets, constraining the second moment of the random disturbance. In the optimal control setting, such moment-based risk measures lead to tractable optimal controllers when combined with affine disturbance feedback. Assumptions on the constraints are given that guarantee recursive feasibility. The resulting control scheme acts as a robust controller when little data is available and converges to the certainty equivalent controller when a large sample count implies high confidence in the estimated second moment. This is illustrated in a numerical experiment.

We introduce two algorithms for nonconvex regularized finite sum minimization, where typical Lipschitz differentiability assumptions are relaxed to the notion of relative smoothness. The first one is a Bregman extension of Finito/MISO, studied for fully nonconvex problems when the sampling is randomized, or under convexity of the nonsmooth term when it is essentially cyclic. The second algorithm is a low-memory variant, in the spirit of SVRG and SARAH, that also allows for fully nonconvex formulations. Our analysis is made remarkably simple by employing a Bregman Moreau envelope as Lyapunov function. In the randomized case, linear convergence is established when the cost function is strongly convex, yet with no convexity requirements on the individual functions in the sum. For the essentially cyclic and low-memory variants, global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property.

Detecting out-of-distribution (OOD) samples is an essential requirement for the deployment of machine learning systems in the real world. Until now, research on energy-based OOD detectors has focused on the softmax confidence score from a pre-trained neural network classifier with access to class labels. In contrast, we propose an unsupervised energy-based OOD detector leveraging the Stiefel-Restricted Kernel Machine (St-RKM). Training requires minimizing an objective function with an autoencoder loss term and the RKM energy where the interconnection matrix lies on the Stiefel manifold. Further, we outline multiple energy function definitions based on the RKM framework and discuss their utility. In the experiments on standard datasets, the proposed method improves over the existing energy-based OOD detectors and deep generative models. Through several ablation studies, we further illustrate the merit of each proposed energy function on the OOD detection performance.

We introduce Constr-DRKM, a deep kernel method for the unsupervised learning of disentangled data representations. We propose augmenting the original deep restricted kernel machine formulation for kernel PCA by orthogonality constraints on the latent variables to promote disentanglement and to make it possible to carry out optimization without first defining a stabilized objective. After illustrating an end-to-end training procedure based on a quadratic penalty optimization algorithm with warm start, we quantitatively evaluate the proposed method's effectiveness in disentangled feature learning. We demonstrate on four benchmark datasets that this approach performs similarly overall to $\beta$-VAE on a number of disentanglement metrics when few training points are available, while being less sensitive to randomness and hyperparameter selection than $\beta$-VAE. We also present a deterministic initialization of Constr-DRKM's training algorithm that significantly improves the reproducibility of the results. Finally, we empirically evaluate and discuss the role of the number of layers in the proposed methodology, examining the influence of each principal component in every layer and showing that components in lower layers act as local feature detectors capturing the broad trends of the data distribution, while components in deeper layers use the representation learned by previous layers and more accurately reproduce higher-level features.

We propose QPALM, a nonconvex quadratic programming (QP) solver based on the proximal augmented Lagrangian method. This method solves a sequence of inner subproblems which can be enforced to be strongly convex and which therefore admit a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extended-real-valued cost function, which allows for a fairly simple analysis where convergence to a stationary point at an \(R\)-linear rate is shown. The QPALM algorithm solves the subproblems iteratively using semismooth Newton directions and an exact linesearch. The former can be computed efficiently in most iterations by making use of suitable factorization update routines, while the latter requires the zero of a monotone, one-dimensional, piecewise affine function. QPALM is implemented in open-source C code, with tailored linear algebra routines for the factorization in a self-written package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the Maros-Meszaros problems and finding a stationary point for most of the nonconvex QPs in the Cutest test set. Furthermore, it is shown to be competitive against state-of-the-art convex QP solvers in typical QPs arising from application domains such as portfolio optimization and model predictive control. As such, QPALM strikes a unique balance between solving both easy and hard problems efficiently.

We present a data-driven model predictive control scheme for chance-constrained Markovian switching systems with unknown switching probabilities. Using samples of the underlying Markov chain, ambiguity sets of transition probabilities are estimated which include the true conditional probability distributions with high probability. These sets are updated online and used to formulate a time-varying, risk-averse optimal control problem. We prove recursive feasibility of the resulting MPC scheme and show that the original chance constraints remain satisfied at every time step. Furthermore, we show that under sufficient decrease of the confidence levels, the resulting MPC scheme renders the closed-loop system mean-square stable with respect to the true-but-unknown distributions, while remaining less conservative than a fully robust approach., Comment: Accepted at 59th IEEE Conference on Decision and Control (CDC 2020); extended version

This paper studies the sample complexity of the stochastic Linear Quadratic Regulator when applied to systems with multiplicative noise. We assume that the covariance of the noise is unknown and estimate it using the sample covariance, which results in suboptimal behaviour. The main contribution of this paper is then to bound the suboptimality of the methodology and prove that it decreases with 1/N, where N denotes the amount of samples. The methodology easily generalizes to the case where the mean is unknown and to the distributionally robust case studied in a previous work of the authors. The analysis is mostly based on results from matrix function perturbation analysis.

Although the performance of popular optimization algorithms such as Douglas-Rachford splitting (DRS) and the ADMM is satisfactory in small and well-scaled problems, ill conditioning and problem size pose a severe obstacle to their reliable employment. Expanding on recent convergence results for DRS and ADMM applied to nonconvex problems, we propose two linesearch algorithms to enhance and robustify these methods by means of quasi-Newton directions. The proposed algorithms are suited for nonconvex problems, require the same black-box oracle of DRS and ADMM, and maintain their (subsequential) convergence properties. Numerical evidence shows that the employment of L-BFGS in the proposed framework greatly improves convergence of DRS and ADMM, making them robust to ill conditioning. Under regularity and nondegeneracy assumptions at the limit point, superlinear convergence is shown when quasi-Newton Broyden directions are adopted.

We propose a learning-based, distributionally robust model predictive control approach towards the design of adaptive cruise control (ACC) systems. We model the preceding vehicle as an autonomous stochastic system, using a hybrid model with continuous dynamics and discrete, Markovian inputs. We estimate the (unknown) transition probabilities of this model empirically using observed mode transitions and simultaneously determine sets of probability vectors (ambiguity sets) around these estimates, that contain the true transition probabilities with high confidence. We then solve a risk-averse optimal control problem that assumes the worst-case distributions in these sets. We furthermore derive a robust terminal constraint set and use it to establish recursive feasibility of the resulting MPC scheme. We validate the theoretical results and demonstrate desirable properties of the scheme through closed-loop simulations., Comment: Accepted for publication in the IFAC World Congress 2020; extended version with proofs

Difference-of-convex (DC) optimization problems are shown to be equivalent to the minimization of a Lipschitz-differentiable "envelope". A gradient method on this surrogate function yields a novel (sub)gradient-free proximal algorithm which is inherently parallelizable and can handle fully nonsmooth formulations. Newton-type methods such as L-BFGS are directly applicable with a classical linesearch. Our analysis reveals a deep kinship between the novel DC envelope and the forward-backward envelope, the former being a smooth and convexity-preserving nonlinear reparametrization of the latter.

We propose BIBPA, a block inertial Bregman proximal algorithm for minimizing the sum of a block relatively smooth function (that is, relatively smooth concerning each block) and block separable nonsmooth nonconvex functions. We prove that the sequence generated by BIBPA subsequentially converges to critical points of the objective under standard assumptions, and globally converges when the objective function is additionally assumed to satisfy the Kurdyka-{\L}ojasiewicz (K{\L}) property. We also provide the convergence rate when the objective satisfies the {\L}ojasiewicz inequality. We apply BIBPA to the symmetric nonnegative matrix tri-factorization (SymTriNMF) problem, where we propose kernel functions for SymTriNMF and provide closed-form solutions for subproblems of BIBPA., Comment: 18 pages

The decomposition of tensors into simple rank-1 terms is key in a variety of applications in signal processing, data analysis and machine learning. While this canonical polyadic decomposition (CPD) is unique under mild conditions, including prior knowledge such as nonnegativity can facilitate interpretation of the components. Inspired by the effectiveness and efficiency of Gauss-Newton (GN) for unconstrained CPD, we derive a proximal, semismooth GN type algorithm for nonnegative tensor factorization. If the algorithm converges to the global optimum, we show that $Q$-quadratic convergence can be obtained in the exact case. Global convergence is achieved via backtracking on the forward-backward envelope function. The $Q$-quadratic convergence is verified experimentally, and we illustrate that using the GN step significantly reduces number of (expensive) gradient computations compared to proximal gradient descent.

We present Optimization Engine (OpEn): an open-source code generation tool for real-time embedded nonconvex optimization, which implements a novel numerical method. OpEn combines the proximal averaged Newton-type method for optimal control (PANOC) with the penalty and augmented Lagrangian methods to compute approximate stationary points of nonconvex problems. The proposed method involves very simple algebraic operations such as vector products, has a low memory footprint and exhibits very good convergence properties that allow the solution of nonconvex problems on embedded devices. OpEn's core solver is written is Rust - a modern, high-performance, memory-safe and thread-safe systems programming language - while users can call it from Python, MATLAB, C, C++ or over a TCP socket., Comment: IFAC World Congress 2020, Berlin

We present a data-driven method for solving the linear quadratic regulator problem for systems with multiplicative disturbances, the distribution of which is only known through sample estimates. We adopt a distributionally robust approach to cast the controller synthesis problem as semidefinite programs. Using results from high dimensional statistics, the proposed methodology ensures that their solution provides mean-square stabilizing controllers with high probability even for low sample sizes. As sample size increases the closed-loop cost approaches that of the optimal controller produced when the distribution is known. We demonstrate the practical applicability and performance of the method through a numerical experiment., Comment: Extended technical report. Accepted for oral presentation at L4DC2020

We present a proximal augmented Lagrangian based solver for general convex quadratic programs (QPs), relying on semismooth Newton iterations with exact line search to solve the inner subproblems. The exact line search reduces in this case to finding the zero of a one-dimensional monotone, piecewise affine function and can be carried out very efficiently. Our algorithm requires the solution of a linear system at every iteration, but as the matrix to be factorized depends on the active constraints, efficient sparse factorization updates can be employed like in active-set methods. Both primal and dual residuals can be enforced down to strict tolerances and otherwise infeasibility can be detected from intermediate iterates. A C implementation of the proposed algorithm is tested and benchmarked against other state-of-the-art QP solvers for a large variety of problem data and shown to compare favorably against these solvers.

We introduce and analyze BPALM and A-BPALM, two multi-block proximal alternating linearized minimization algorithms using Bregman distances for solving structured nonconvex problems. The objective function is the sum of a multi-block relatively smooth function (i.e., relatively smooth by fixing all the blocks except one) and block separable (nonsmooth) nonconvex functions. It turns out that the sequences generated by our algorithms are subsequentially convergent to critical points of the objective function, while they are globally convergent under KL inequality assumption. Further, the rate of convergence is further analyzed for functions satisfying the {\L}ojasiewicz's gradient inequality. We apply this framework to orthogonal nonnegative matrix factorization (ONMF) that satisfies all of our assumptions and the related subproblems are solved in closed forms, where some preliminary numerical results is reported.

This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies. This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies.

We introduce Bella, a locally superlinearly convergent Bregman forward backward splitting method for minimizing the sum of two nonconvex functions, one of which satisfying a relative smoothness condition and the other one possibly nonsmooth. A key tool of our methodology is the Bregman forward-backward envelope (BFBE), an exact and continuous penalty function with favorable first- and second-order properties, and enjoying a nonlinear error bound when the objective function satisfies a Lojasiewicz-type property. The proposed algorithm is of linesearch type over the BFBE along candidate update directions, and converges subsequentially to stationary points, globally under a KL condition, and owing to the given nonlinear error bound can attain superlinear convergence rates even when the limit point is a nonisolated minimum, provided the directions are suitably selected.

We employ the proximal averaged Newton-type method for optimal control (PANOC) to solve obstacle avoidance problems in real time. We introduce a novel modeling framework for obstacle avoidance which allows us to easily account for generic, possibly nonconvex, obstacles involving polytopes, ellipsoids, semialgebraic sets and generic sets described by a set of nonlinear inequalities. PANOC is particularly well-suited for embedded applications as it involves simple steps, its implementation comes with a low memory footprint and its fast convergence meets the tight runtime requirements of fast dynamical systems one encounters in modern mechatronics and robotics. The proposed obstacle avoidance scheme is tested on a lab-scale autonomous vehicle.

We present an open-source solution for the operational control of drinking water distribution networks which accounts for the inherent uncertainty in water demand and electricity prices in the day-ahead market of a volatile deregulated economy. As increasingly more energy markets adopt this trading scheme, the operation of drinking water networks requires uncertainty-aware control approaches that mitigate the effect of volatility and result in an economic and safe operation of the network that meets the consumers' need for uninterrupted water supply. We propose the use of scenario-based stochastic model predictive control: an advanced control methodology which comes at a considerable computation cost which is overcome by harnessing the parallelization capabilities of graphics processing units (GPUs) and using a massively parallelizable algorithm based on the accelerated proximal gradient method., Comment: Drinking water networks, Bid-based energy market, Stochastic model predictive control, Graphics processing units, Scenario trees, Open-source software

The coordination of highly automated vehicles (or agents) in road intersections is an inherently nonconvex and challenging problem. In this paper, we propose a distributed motion planning scheme under reasonable vehicle-to-vehicle communication requirements. Each agent solves a nonlinear model predictive control problem in real time and transmits its planned trajectory to other agents, which may have conflicting objectives. The problem formulation is augmented with conditional constraints that enable the agents to decide whether to wait at a stopping line, if safe crossing is not possible. The involved nonconvex problems are solved very efficiently using the proximal averaged Newton method for optimal control (PANOC). We demonstrate the efficiency of the proposed approach in a realistic intersection crossing scenario., Comment: IEEE Conference on Decision and Control (CDC) 2019

We consider the problem of designing control laws for stochastic jump linear systems where the disturbances are drawn randomly from a finite sample space according to an unknown distribution, which is estimated from a finite sample of i.i.d. observations. We adopt a distributionally robust approach to compute a mean-square stabilizing feedback gain with a given probability. The larger the sample size, the less conservative the controller, yet our methodology gives stability guarantees with high probability, for any number of samples. Using tools from statistical learning theory, we estimate confidence regions for the unknown probability distributions (ambiguity sets) which have the shape of total variation balls centered around the empirical distribution. We use these confidence regions in the design of appropriate distributionally robust controllers and show that the associated stability conditions can be cast as a tractable linear matrix inequality (LMI) by using conjugate duality. The resulting design procedure scales gracefully with the size of the probability space and the system dimensions. Through a numerical example, we illustrate the superior sample complexity of the proposed methodology over the stochastic approach., Comment: Accepted to the 2019 IEEE Conference on Decision and Control (CDC)

Multistage risk-averse optimal control problems with nested conditional risk mappings are gaining popularity in various application domains. Risk-averse formulations interpolate between the classical expectation-based stochastic and minimax optimal control. This way, risk-averse problems aim at hedging against extreme low-probability events without being overly conservative. At the same time, risk-based constraints may be employed either as surrogates for chance (probabilistic) constraints or as a robustification of expectation-based constraints. Such multistage problems, however, have been identified as particularly hard to solve. We propose a decomposition method for such nested problems that allows us to solve them via efficient numerical optimization methods. Alongside, we propose a new form of risk constraints which accounts for the propagation of uncertainty in time., Comment: Please, cite this work as P. Sopasakis, M. Schuurmans, P. Patrinos, "Risk-averse risk-constrained optimal control," IEEE ECC, Naples, Italy, 2019

We present SuperSCS: a fast and accurate method for solving large-scale convex conic problems. SuperSCS combines the SuperMann algorithmic framework with the Douglas-Rachford splitting which is applied on the homogeneous self-dual embedding of conic optimization problems: a model for conic optimization problems which simultaneously encodes the optimality conditions and infeasibility/unboundedness certificates for the original problem. SuperMann allows the use of fast quasi-Newtonian directions such as a modified restarted Broyden-type direction and Anderson's acceleration., Comment: Cite as: P. Sopasakis, K. Menounou, P. Patrinos, "SuperSCS: fast and accurate large-scale conic optimization," IEEE European Control Conference, Naples, Italy, 2019

We propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. Our methods possess three main advantages compared to current state-of-the-art accelerated first-order methods: (1) they allow using two different extrapolation points to evaluate the gradients and to add the inertial force (we will empirically show that it is more efficient than using a single extrapolation point), (2) they allow to randomly picking the block of variables to update, and (3) they do not require a restarting step. We prove the subsequential convergence of the generated sequence under mild assumptions, prove the global convergence under some additional assumptions, and provide convergence rates. We deploy the proposed methods to solve non-negative matrix factorization (NMF) and show that they compete favorably with the state-of-the-art NMF algorithms. Additional experiments on non-negative approximate canonical polyadic decomposition, also known as non-negative tensor factorization, are also provided.

We propose a methodology for autonomous aerial navigation and obstacle avoidance of micro aerial vehicles (MAV) using nonlinear model predictive control (NMPC) and we demonstrate its effectiveness with laboratory experiments. The proposed methodology can accommodate obstacles of arbitrary, potentially non-convex, geometry. The NMPC problem is solved using PANOC: a fast numerical optimization method which is completely matrix-free, is not sensitive to ill conditioning, involves only simple algebraic operations and is suitable for embedded NMPC. A C89 implementation of PANOC solves the NMPC problem at a rate of 20Hz on board a lab-scale MAV. The MAV performs smooth maneuvers moving around an obstacle. For increased autonomy, we propose a simple method to compensate for the reduction of thrust over time, which comes from the depletion of the MAV's battery, by estimating the thrust constant.

We propose a Forward-Backward Truncated-Newton method (FBTN) for minimizing the sum of two convex functions, one of which smooth. Unlike other proximal Newton methods, our approach does not involve the employment of variable metrics, but is rather based on a reformulation of the original problem as the unconstrained minimization of a continuously differentiable function, the forward-backward envelope (FBE). We introduce a generalized Hessian for the FBE that symmetrizes the generalized Jacobian of the nonlinear system of equations representing the optimality conditions for the problem. This enables the employment of conjugate gradient method (CG) for efficiently solving the resulting (regularized) linear systems, which can be done inexactly. The employment of CG prevents the computation of full (generalized) Jacobians, as it requires only (generalized) directional derivatives. The resulting algorithm is globally (subsequentially) convergent, $Q$-linearly under an error bound condition, and up to $Q$-superlinearly and $Q$-quadratically under regularity assumptions at the possibly non-isolated limit point.

We consider algorithms for solving structured convex optimization problems over a network of agents with communication delays. It is assumed that each agent performs its local updates by using possibly outdated information from its neighbors under the assumption that the delay with respect to each neighbor is bounded but otherwise arbitrary. The private objective of each agent is represented by the sum of two possibly nonsmooth functions, one of which is composed with a linear mapping. The global optimization problem is the aggregate of the local cost functions and a common Lipschitz-differentiable term. When the coupling between the agents is represented only through the common function the primal-dual algorithm proposed by V\~u and Condat can be conveniently employed, while for more general structures a new algorithm is proposed. Moreover, a randomized variant is presented that allows the agents to wake up at random and independently from one another. The convergence of each of the proposed algorithms is established under different strong convexity assumptions.

In this paper we present a risk-averse model predictive control (MPC) scheme for the operation of islanded microgrids with very high share of renewable energy sources. The proposed scheme mitigates the effect of errors in the determination of the probability distribution of renewable infeed and load. This allows to use less complex and less accurate forecasting methods and to formulate low-dimensional scenario-based optimisation problems which are suitable for control applications. Additionally, the designer may trade performance for safety by interpolating between the conventional stochastic and worst-case MPC formulations. The presented risk-averse MPC problem is formulated as a mixed-integer quadratically-constrained quadratic problem and its favourable characteristics are demonstrated in a case study. This includes a sensitivity analysis that illustrates the robustness to load and renewable power prediction errors.

This paper presents a novel model predictive control strategy for controlling autonomous motion systems moving through an environment with obstacles of general shape. In order to solve such a generic non-convex optimization problem and find a feasible trajectory that reaches the destination, the approach employs a quadratic penalty method to enforce the obstacle avoidance constraints, and several heuristics to bypass local minima behind an obstacle. The quadratic penalty method itself aids in avoiding such local minima by gradually finding a path around the obstacle as the penalty factors are successively increased. The inner optimization problems are solved in real time using the proximal averaged Newton-type method for optimal control (PANOC), a first-order method which exhibits low runtime and is suited for embedded applications. The method is validated by extensive numerical simulations and shown to outperform state-of-the-art solvers in runtime and robustness., Comment: 7 pages