arXiv subjects

Mathias Staudigl

Publications and source records attributed to Mathias Staudigl.

At least 37 records · Page 2Linked to original sources

arXiv2023

Random block coordinate methods for inconsistent convex optimisation problems

We develop a novel randomised block coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying in the midway between the celebrated Chambolle-Pock primal-dual algorithm and Tseng's accelerated proximal gradient method, we establish global convergence of the last iterate as well optimal $O(1/k)$ and $O(1/k^{2})$ complexity rates in the convex and strongly convex case, respectively, $k$ being the iteration count. Motivated by the increased complexity in the control of distribution level electric power systems, we test the performance of our method on a second-order cone relaxation of an AC-OPF problem. Distributed control is achieved via the distributed locational marginal prices (DLMPs), which are obtained \revise{as} dual variables in our optimisation framework.

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arXiv2022

Proximal-like algorithms for equilibrium seeking in mixed-integer Nash equilibrium problems

We consider potential games with mixed-integer variables, for which we propose two distributed, proximal-like equilibrium seeking algorithms. Specifically, we focus on two scenarios: i) the underlying game is generalized ordinal and the agents update through iterations by choosing an exact optimal strategy; ii) the game admits an exact potential and the agents adopt approximated optimal responses. By exploiting the properties of integer-compatible regularization functions used as penalty terms, we show that both algorithms converge to either an exact or an $ε$-approximate equilibrium. We corroborate our findings on a numerical instance of a Cournot oligopoly model.

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arXiv2022

Hessian barrier algorithms for non-convex conic optimization

We consider the minimization of a continuous function over the intersection of a regular cone with an affine set via a new class of adaptive first- and second-order optimization methods, building on the Hessian-barrier techniques introduced in [Bomze, Mertikopoulos, Schachinger, and Staudigl, Hessian barrier algorithms for linearly constrained optimization problems, SIAM Journal on Optimization, 2019]. Our approach is based on a potential-reduction mechanism and attains a suitably defined class of approximate first- or second-order KKT points with the optimal worst-case iteration complexity $O(\varepsilon^{-2})$ (first-order) and $O(\varepsilon^{-3/2})$ (second-order), respectively. A key feature of our methodology is the use of self-concordant barrier functions to construct strictly feasible iterates via a disciplined decomposition approach and without sacrificing on the iteration complexity of the method. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained optimization problems.

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arXiv2022

A privacy-preserving distributed computational approach for distributed locational marginal prices

An important issue in today's electricity markets is the management of flexibilities offered by new practices, such as smart home appliances or electric vehicles. By inducing changes in the behavior of residential electric utilities, demand response (DR) seeks to adjust the demand of power to the supply for increased grid stability and better integration of renewable energies. A key role in DR is played by emergent independent entities called load aggregators (LAs). We develop a new decentralized algorithm to solve a convex relaxation of the classical Alternative Current Optimal Power Flow (ACOPF) problem, which relies on local information only. Each computational step can be performed in an entirely privacy-preserving manner, and system-wide coordination is achieved via node-specific distribution locational marginal prices (DLMPs). We demonstrate the efficiency of our approach on a 15-bus radial distribution network.

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arXiv2022

Mini-batch stochastic three-operator splitting for distributed optimization

We consider a network of agents, each with its own private cost consisting of a sum of two possibly nonsmooth convex functions, one of which is composed with a linear operator. At every iteration each agent performs local calculations and can only communicate with its neighbors. The challenging aspect of our study is that the smooth part of the private cost function is given as an expected value and agents only have access to this part of the problem formulation via a heavy-tailed stochastic oracle. To tackle such sampling-based optimization problems, we propose a stochastic extension of the triangular pre-conditioned primal-dual algorithm. We demonstrate almost sure convergence of the scheme and validate the performance of the method via numerical experiments.

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arXiv2021

Multi-agent online learning in time-varying games

We examine the long-run behavior of multi-agent online learning in games that evolve over time. Specifically, we focus on a wide class of policies based on mirror descent, and we show that the induced sequence of play (a) converges to Nash equilibrium in time-varying games that stabilize in the long run to a strictly monotone limit; and (b) it stays asymptotically close to the evolving equilibrium of the sequence of stage games (assuming they are strongly monotone). Our results apply to both gradient-based and payoff-based feedback - i.e., the "bandit feedback" case where players only get to observe the payoffs of their chosen actions.

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arXiv2021

Stochastic Relaxed Inertial Forward-Backward-Forward splitting for Monotone Inclusions in Hilbert spaces

We consider monotone inclusions defined on a Hilbert space where the operator is given by the sum of a maximal monotone operator $T$ and a single-valued monotone, Lipschitz continuous, and expectation-valued operator $V$. We draw motivation from the seminal work by Attouch and Cabot on relaxed inertial methods for monotone inclusions and present a stochastic extension of the relaxed inertial forward-backward-forward (RISFBF) method. Facilitated by an online variance reduction strategy via a mini-batch approach, we show that (RISFBF) produces a sequence that weakly converges to the solution set. Moreover, it is possible to estimate the rate at which the discrete velocity of the stochastic process vanishes. Under strong monotonicity, we demonstrate strong convergence, and give a detailed assessment of the iteration and oracle complexity of the scheme. When the mini-batch is raised at a geometric (polynomial) rate, the rate statement can be strengthened to a linear (suitable polynomial) rate while the oracle complexity of computing an $ε$-solution improves to $O(1/ε)$. Importantly, the latter claim allows for possibly biased oracles, a key theoretical advancement allowing for far broader applicability. By defining a restricted gap function based on the Fitzpatrick function, we prove that the expected gap of an averaged sequence diminishes at a sublinear rate of $O(1/k)$ while the oracle complexity of computing a suitably defined $ε$-solution is $O(1/ε^{1+a})$ where $a>1$. Numerical results on two-stage games and an overlapping group Lasso problem illustrate the advantages of our method compared to stochastic forward-backward-forward (SFBF) and SA schemes.

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arXiv2021

Generalized Self-Concordant Analysis of Frank-Wolfe algorithms

Projection-free optimization via different variants of the Frank-Wolfe (FW) method has become one of the cornerstones in large scale optimization for machine learning and computational statistics. Numerous applications within these fields involve the minimization of functions with self-concordance like properties. Such generalized self-concordant (GSC) functions do not necessarily feature a Lipschitz continuous gradient, nor are they strongly convex. Indeed, in a number of applications, e.g. inverse covariance estimation or distance-weighted discrimination problems in support vector machines, the loss is given by a GSC function having unbounded curvature, implying absence of theoretical guarantees for the existing FW methods. This paper closes this apparent gap in the literature by developing provably convergent FW algorithms with standard O(1/k) convergence rate guarantees. If the problem formulation allows the efficient construction of a local linear minimization oracle, we develop a FW method with linear convergence rate.

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arXiv2021

Computing Dynamic User Equilibrium on Large-Scale Networks Without Knowing Global Parameters

Dynamic user equilibrium (DUE) is a Nash-like solution concept describing an equilibrium in dynamic traffic systems over a fixed planning period. DUE is a challenging class of equilibrium problems, connecting network loading models and notions of system equilibrium in one concise mathematical framework. Recently, Friesz and Han introduced an integrated framework for DUE computation on large-scale networks, featuring a basic fixed-point algorithm for the effective computation of DUE. In the same work, they present an open-source MATLAB toolbox which allows researchers to test and validate new numerical solvers. This paper builds on this seminal contribution, and extends it in several important ways. At a conceptual level, we provide new strongly convergent algorithms designed to compute a DUE directly in the infinite-dimensional space of path flows. An important feature of our algorithms is that they give provable convergence guarantees without knowledge of global parameters. In fact, the algorithms we propose are adaptive, in the sense that they do not need a priori knowledge of global parameters of the delay operator, and which are provable convergent even for delay operators which are non-monotone. We implement our numerical schemes on standard test instances, and compare them with the numerical solution strategy employed by Friesz and Han.

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arXiv2021

A relaxed-inertial forward-backward-forward algorithm for Stochastic Generalized Nash equilibrium seeking

In this paper we propose a new operator splitting algorithm for distributed Nash equilibrium seeking under stochastic uncertainty, featuring relaxation and inertial effects. Our work is inspired by recent deterministic operator splitting methods, designed for solving structured monotone inclusion problems. The algorithm is derived from a forward-backward-forward scheme for solving structured monotone inclusion problems featuring a Lipschitz continuous and monotone game operator. To the best of our knowledge, this is the first distributed (generalized) Nash equilibrium seeking algorithm featuring acceleration techniques in stochastic Nash games without assuming cocoercivity. Numerical examples illustrate the effect of inertia and relaxation on the performance of our proposed algorithm.

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arXiv2021

First-Order Methods for Convex Optimization

First-order methods for solving convex optimization problems have been at the forefront of mathematical optimization in the last 20 years. The rapid development of this important class of algorithms is motivated by the success stories reported in various applications, including most importantly machine learning, signal processing, imaging and control theory. First-order methods have the potential to provide low accuracy solutions at low computational complexity which makes them an attractive set of tools in large-scale optimization problems. In this survey we cover a number of key developments in gradient-based optimization methods. This includes non-Euclidean extensions of the classical proximal gradient method, and its accelerated versions. Additionally we survey recent developments within the class of projection-free methods, and proximal versions of primal-dual schemes. We give complete proofs for various key results, and highlight the unifying aspects of several optimization algorithms.

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arXiv2020

A competitive search game with a moving target

We introduce a discrete-time search game, in which two players compete to find an object first. The object moves according to a time-varying Markov chain on finitely many states. The players know the Markov chain and the initial probability distribution of the object, but do not observe the current state of the object. The players are active in turns. The active player chooses a state, and this choice is observed by the other player. If the object is in the chosen state, this player wins and the game ends. Otherwise, the object moves according to the Markov chain and the game continues at the next period. We show that this game admits a value, and for any error-term $\veps>0$, each player has a pure (subgame-perfect) $\veps$-optimal strategy. Interestingly, a 0-optimal strategy does not always exist. The $\veps$-optimal strategies are robust in the sense that they are $2\veps$-optimal on all finite but sufficiently long horizons, and also $2\veps$-optimal in the discounted version of the game provided that the discount factor is close to 1. We derive results on the analytic and structural properties of the value and the $\veps$-optimal strategies. Moreover, we examine the performance of the finite truncation strategies, which are easy to calculate and to implement. We devote special attention to the important time-homogeneous case, where additional results hold.

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arXiv2020

Self-Concordant Analysis of Frank-Wolfe Algorithms

Projection-free optimization via different variants of the Frank-Wolfe (FW), a.k.a. Conditional Gradient method has become one of the cornerstones in optimization for machine learning since in many cases the linear minimization oracle is much cheaper to implement than projections and some sparsity needs to be preserved. In a number of applications, e.g. Poisson inverse problems or quantum state tomography, the loss is given by a self-concordant (SC) function having unbounded curvature, implying absence of theoretical guarantees for the existing FW methods. We use the theory of SC functions to provide a new adaptive step size for FW methods and prove global convergence rate O(1/k) after k iterations. If the problem admits a stronger local linear minimization oracle, we construct a novel FW method with linear convergence rate for SC functions.

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arXiv2020

Incentive compatibility in sender-receiver stopping games

We introduce a model of sender-receiver stopping games, where the state of the world follows an iid--process throughout the game. At each period, the sender observes the current state, and sends a message to the receiver, suggesting either to stop or to continue. The receiver, only seeing the message but not the state, decides either to stop the game, or to continue which takes the game to the next period. The payoff to each player is a function of the state when the receiver quits, with higher states leading to better payoffs. The horizon of the game can be finite or infinite. We prove existence and uniqueness of responsive (i.e. non-babbling) Perfect Bayesian Equilibrium (PBE) under mild conditions on the game primitives in the case where the players are sufficiently patient. The responsive PBE has a remarkably simple structure, which builds on the identification of an easy-to-implement and compute class of threshold strategies for the sender. With the help of these threshold strategies, we derive simple expressions describing this PBE. It turns out that in this PBE the receiver obediently follows the recommendations of the sender. Hence, surprisingly, the sender alone plays the decisive role, and regardless of the payoff function of the receiver the sender always obtains the best possible payoff for himself.

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arXiv2020

Distributed forward-backward (half) forward algorithms for generalized Nash equilibrium seeking

We present two distributed algorithms for the computation of a generalized Nash equilibrium in monotone games. The first algorithm follows from a forward-backward-forward operator splitting, while the second, which requires the pseudo-gradient mapping of the game to be cocoercive, follows from the forward-backward-half-forward operator splitting. Finally, we compare them with the distributed, preconditioned, forward-backward algorithm via numerical experiments.

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arXiv2019

Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskii-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

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arXiv2019

Generalized Self-concordant Hessian-barrier algorithms

Many problems in statistical learning, imaging, and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm recently introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt. 2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a generalized self-concordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in the literature of interior point methods. We prove global convergence to an approximate stationary point of the method, and in cases where the feasible set admits an easily computable self-concordant barrier, we verify worst-case optimal iteration complexity of the method. Applications in non-convex statistical estimation and $L^{p}$-minimization are discussed to given the efficiency of the method.

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arXiv2019

Strong Convergence of Forward-Backward-Forward Methods for Pseudo-monotone Variational Inequalities with Applications to Dynamic User Equilibrium in Traffic Networks

In infinite-dimensional Hilbert spaces we device a class of strongly convergent primal-dual schemes for solving variational inequalities defined by a Lipschitz continuous and pseudomonote map. Our novel numerical scheme is based on Tseng's forward-backward-forward scheme, which is known to display weak convergence, unless very strong global monotonicity assumptions are made on the involved operators. We provide a simple augmentation of this algorithm which is computationally cheap and still guarantees strong convergence to a minimal norm solution of the underlying problem. We provide an adaptive extension of the algorithm, freeing us from requiring knowledge of the global Lipschitz constant. We test the performance of the algorithm in the computationally challenging task to find dynamic user equilibria in traffic networks and verify that our scheme is at least competitive to state-of-the-art solvers, and in some case even improve upon them.

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