arXiv subjects

Panagiotis Patrinos

Publications and source records attributed to Panagiotis Patrinos.

At least 37 records · Page 2Linked to original sources

arXiv2025

The inexact power augmented Lagrangian method for constrained nonconvex optimization

This work introduces an unconventional inexact augmented Lagrangian method where the augmenting term is a Euclidean norm raised to a power between one and two. The proposed algorithm is applicable to a broad class of constrained nonconvex minimization problems that involve nonlinear equality constraints. In a first part of this work, we conduct a full complexity analysis of the method under a mild regularity condition, leveraging an accelerated first-order algorithm for solving the Hölder-smooth subproblems. Interestingly, this worst-case result indicates that using lower powers for the augmenting term leads to faster constraint satisfaction, albeit with a slower decrease of the dual residual. Notably, our analysis does not assume boundedness of the iterates. Thereafter, we present an inexact proximal point method for solving the weakly-convex and Hölder-smooth subproblems, and demonstrate that the combined scheme attains an improved rate that reduces to the best-known convergence rate whenever the augmenting term is a classical squared Euclidean norm. Different augmenting terms, involving a lower power, further improve the primal complexity at the cost of the dual complexity. Finally, numerical experiments validate the practical performance of unconventional augmenting terms.

math.OC↗

arXiv2025

ExAMPC: the Data-Driven Explainable and Approximate NMPC with Physical Insights

Amidst the surge in the use of Artificial Intelligence (AI) for control purposes, classical and model-based control methods maintain their popularity due to their transparency and deterministic nature. However, advanced controllers like Nonlinear Model Predictive Control (NMPC), despite proven capabilities, face adoption challenges due to their computational complexity and unpredictable closed-loop performance in complex validation systems. This paper introduces ExAMPC, a methodology bridging classical control and explainable AI by augmenting the NMPC with data-driven insights to improve the trustworthiness and reveal the optimization solution and closed-loop performance's sensitivities to physical variables and system parameters. By employing a low-order spline embedding, we reduce the open-loop trajectory dimensionality by over 95%, and integrate it with SHAP and Symbolic Regression from eXplainable AI (XAI) for an approximate NMPC, enabling intuitive physical insights into the NMPC's optimization routine. The prediction accuracy of the approximate NMPC is enhanced through physics-inspired continuous-time constraints penalties, reducing the predicted continuous trajectory violations by 93%. ExAMPC also enables accurate forecasting of the NMPC's computational requirements with explainable insights on worst-case scenarios. Experimental validation on automated valet parking and autonomous racing with lap-time optimization, demonstrates the methodology's practical effectiveness for potential real-world applications.

eess.SY↗

arXiv2025

Rethinking PCA Through Duality

Motivated by the recently shown connection between self-attention and (kernel) principal component analysis (PCA), we revisit the fundamentals of PCA. Using the difference-of-convex (DC) framework, we present several novel formulations and provide new theoretical insights. In particular, we show the kernelizability and out-of-sample applicability for a PCA-like family of problems. Moreover, we uncover that simultaneous iteration, which is connected to the classical QR algorithm, is an instance of the difference-of-convex algorithm (DCA), offering an optimization perspective on this longstanding method. Further, we describe new algorithms for PCA and empirically compare them with state-of-the-art methods. Lastly, we introduce a kernelizable dual formulation for a robust variant of PCA that minimizes the $l_1$ deviation of the reconstruction errors.

cs.LG↗

arXiv2025

Scaled Relative Graphs for Nonmonotone Operators with Applications in Circuit Theory

The scaled relative graph (SRG) is a powerful graphical tool for analyzing the properties of operators, by mapping their graph onto the complex plane. In this work, we study the SRG of two classes of nonmonotone operators, namely the general class of semimonotone operators and a class of angle-bounded operators. In particular, we provide an analytical description of the SRG of these classes and show that membership of an operator to these classes can be verified through geometric containment of its SRG. To illustrate the importance of these results, we provide several examples in the context of electrical circuits. Most notably, we show that the Ebers-Moll transistor belongs to the class of angle-bounded operators and use this result to compute the response of a common-emitter amplifier using Chambolle-Pock, despite the underlying nonsmoothness and multi-valuedness, leveraging recent convergence results for this algorithm in the nonmonotone setting.

math.OC↗

arXiv2025

Nonlinearly Preconditioned Gradient Methods: Momentum and Stochastic Analysis

We study nonlinearly preconditioned gradient methods for smooth nonconvex optimization problems, focusing on sigmoid preconditioners that inherently perform a form of gradient clipping akin to the widely used gradient clipping technique. Building upon this idea, we introduce a novel heavy ball-type algorithm and provide convergence guarantees under a generalized smoothness condition that is less restrictive than traditional Lipschitz smoothness, thus covering a broader class of functions. Additionally, we develop a stochastic variant of the base method and study its convergence properties under different noise assumptions. We compare the proposed algorithms with baseline methods on diverse tasks from machine learning including neural network training.

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arXiv2025

Global Convergence Analysis of the Power Proximal Point and Augmented Lagrangian Method

In this paper we study an unconventional inexact Augmented Lagrangian Method (ALM) for convex optimization problems, as first proposed by Bertsekas, wherein the penalty term is a potentially non-Euclidean norm raised to a power between one and two. We analyze the algorithm through the lens of a nonlinear Proximal Point Method (PPM), as originally introduced by Luque, applied to the dual problem. While Luque analyzes the order of local convergence of the scheme with Euclidean norms our focus is on the non-Euclidean case which prevents us from using standard tools for the analysis such as the nonexpansiveness of the proximal mapping. To allow for errors in the primal update, we derive two implementable stopping criteria under which we analyze both the global and the local convergence rates of the algorithm. More specifically, we show that the method enjoys a fast sublinear global rate in general and a local superlinear rate under suitable growth assumptions. We also highlight that the power ALM can be interpreted as classical ALM with an implicitly defined penalty-parameter schedule, reducing its parameter dependence. Our experiments on a number of relevant problems suggest that for certain powers the method performs similarly to a classical ALM with fine-tuned adaptive penalty rule, despite involving fewer parameters.

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arXiv2025

Escaping saddle points without Lipschitz smoothness: the power of nonlinear preconditioning

We study generalized smoothness in nonconvex optimization, focusing on $(L_0, L_1)$-smoothness and anisotropic smoothness. The former was empirically derived from practical neural network training examples, while the latter arises naturally in the analysis of nonlinearly preconditioned gradient methods. We introduce a new sufficient condition that encompasses both notions, reveals their close connection, and holds in key applications such as phase retrieval and matrix factorization. Leveraging tools from dynamical systems theory, we then show that nonlinear preconditioning -- including gradient clipping -- preserves the saddle point avoidance property of classical gradient descent. Crucially, the assumptions required for this analysis are actually satisfied in these applications, unlike in classical results that rely on restrictive Lipschitz smoothness conditions. We further analyze a perturbed variant that efficiently attains second-order stationarity with only logarithmic dependence on dimension, matching similar guarantees of classical gradient methods.

math.OC↗

arXiv2025

A Deep Learning Based Resource Allocator for Communication Networks with Dynamic User Utility Demands

Deep learning (DL) based resource allocation (RA) has recently gained significant attention due to its performance efficiency. However, most related studies assume an ideal case where the number of users and their utility demands, e.g., data rate constraints, are fixed, and the designed DL-based RA scheme exploits a policy trained only for these fixed parameters. Consequently, computationally complex policy retraining is required whenever these parameters change. In this paper, we introduce a DL-based resource allocator (ALCOR) that allows users to adjust their utility demands freely, such as based on their application layer requirements. ALCOR employs deep neural networks (DNNs) as the policy in a time-sharing problem. The underlying optimization algorithm iteratively optimizes the on-off status of users to satisfy their utility demands in expectation. The policy performs unconstrained RA (URA) -- RA without considering user utility demands -- among active users to maximize the sum utility (SU) at each time instant. Depending on the chosen URA scheme, ALCOR can perform RA in either a centralized or distributed scenario. The derived convergence analyses provide theoretical guarantees for ALCOR's convergence, and numerical experiments corroborate its effectiveness compared to meta-learning and reinforcement learning approaches.

eess.SP↗

arXiv2025

Second-order methods for provably escaping strict saddle points in composite nonconvex and nonsmooth optimization

This study introduces two second-order methods designed to provably avoid saddle points in composite nonconvex optimization problems: (i) a nonsmooth trust-region method and (ii) a curvilinear linesearch method. These developments are grounded in the forward-backward envelope (FBE), for which we analyze the local second-order differentiability around critical points and establish a novel equivalence between its second-order stationary points and those of the original objective. We show that the proposed algorithms converge to second-order stationary points of the FBE under a mild local smoothness condition on the proximal mapping of the nonsmooth term. Notably, for $ \C^2 $-partly smooth functions, this condition holds under a standard strict complementarity assumption. To the best of our knowledge, these are the first second-order algorithms that provably escape nonsmooth strict saddle points of composite nonconvex optimization, regardless of the initialization. Our preliminary numerical experiments show promising performance of the developed methods, validating our theoretical foundations.

math.OC↗

arXiv2025

Nonlinearly Preconditioned Gradient Methods under Generalized Smoothness

We analyze nonlinearly preconditioned gradient methods for solving smooth minimization problems. We introduce a generalized smoothness property, based on the notion of abstract convexity, that is broader than Lipschitz smoothness and provide sufficient first- and second-order conditions. Notably, our framework encapsulates algorithms associated with the gradient clipping method and brings out novel insights for the class of $(L_0,L_1)$-smooth functions that has received widespread interest recently, thus allowing us to extend beyond already established methods. We investigate the convergence of the proposed method in both the convex and nonconvex setting.

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arXiv2025

Tight Convergence Rates in Gradient Mapping for the Difference-of-Convex Algorithm

We establish new theoretical convergence guarantees for the difference-of-convex algorithm (DCA), where the second function is allowed to be weakly-convex, measuring progress via composite gradient mapping. Based on a tight analysis of two iterations of DCA, we identify six parameter regimes leading to sublinear convergence rates toward critical points and establish those rates by proving adapted descent lemmas. We recover existing rates for the standard difference-of-convex decompositions of nonconvex-nonconcave functions, while for all other curvature settings our results are new, complementing recently obtained rates on the gradient residual. Three of our sublinear rates are tight for any number of DCA iterations, while for the other three regimes we conjecture exact rates, using insights from the tight analysis of gradient descent and numerical validation using the performance estimation methodology. Finally, we show how the equivalence between proximal gradient descent (PGD) and DCA allows the derivation of exact PGD rates for any constant stepsize.

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arXiv2025

Risk-Sensitive Model Predictive Control for Interaction-Aware Planning -- A Sequential Convexification Algorithm

This paper considers risk-sensitive model predictive control for stochastic systems with a decision-dependent distribution. This class of systems is commonly found in human-robot interaction scenarios. We derive computationally tractable convex upper bounds to both the objective function, and to frequently used penalty terms for collision avoidance, allowing us to efficiently solve the generally nonconvex optimal control problem as a sequence of convex problems. Simulations of a robot navigating a corridor demonstrate the effectiveness and the computational advantage of the proposed approach.

math.OC↗

arXiv2025

Optimal Intraday Power Trading for Single-Price Balancing Markets: An Adaptive Risk-Averse Strategy using Mixture Models

Efficient markets are characterised by profit-driven participants continuously refining their positions towards the latest insights. Margins for profit generation are generally small, shaping a difficult landscape for automated trading strategies. This paper introduces a novel intraday power trading strategy tailored for single-price balancing markets. The strategy relies on a strategically devised mixture model to forecast future system imbalance prices and is formulated as a stochastic optimization problem with decision-dependent distributions to address two primary challenges: (i) the impact of trading positions on the system imbalance price and (ii) the uncertainty inherent in the model. The first challenge is tackled by adjusting the model to account for price changes after taking a position. For the second challenge, a coherent risk measure is added to the cost function to take additional uncertainties into account. This paper introduces a methodology to select the tuning parameter of this risk measure adaptively by continuously quantifying the performance of the strategy on a window of recently observed data. The strategy is validated with a simulation on the Belgian electricity market using real-time market data. The adaptive tuning approach leads to higher absolute profits, while also reducing the number of trades.

cs.CE↗

arXiv2025

Spingarn's Method and Progressive Decoupling Beyond Elicitable Monotonicity

Spingarn's method of partial inverses and the progressive decoupling algorithm address inclusion problems involving the sum of an operator and the normal cone of a linear subspace, known as linkage problems. Despite their success, existing convergence results are limited to the so-called elicitable monotone setting, where nonmonotonicity is allowed only on the orthogonal complement of the linkage subspace. In this paper, we introduce progressive decoupling+, a generalized version of standard progressive decoupling that incorporates separate relaxation parameters for the linkage subspace and its orthogonal complement. We prove convergence under conditions that link the relaxation parameters to the nonmonotonicity of their respective subspaces and show that the special cases of Spingarn's method and standard progressive decoupling also extend beyond the elicitable monotone setting. Our analysis hinges upon an equivalence between progressive decoupling+ and the preconditioned proximal point algorithm, for which we develop a general local convergence analysis in a certain nonmonotone setting.

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arXiv2025

Quantization-aware Matrix Factorization for Low Bit Rate Image Compression

Lossy image compression is essential for efficient transmission and storage. Traditional compression methods mainly rely on discrete cosine transform (DCT) or singular value decomposition (SVD), both of which represent image data in continuous domains and, therefore, necessitate carefully designed quantizers. Notably, these methods consider quantization as a separate step, where quantization errors cannot be incorporated into the compression process. The sensitivity of these methods, especially SVD-based ones, to quantization errors significantly degrades reconstruction quality. To address this issue, we introduce a quantization-aware matrix factorization (QMF) to develop a novel lossy image compression method. QMF provides a low-rank representation of the image data as a product of two smaller factor matrices, with elements constrained to bounded integer values, thereby effectively integrating quantization with low-rank approximation. We propose an efficient, provably convergent iterative algorithm for QMF using a block coordinate descent (BCD) scheme, with subproblems having closed-form solutions. Our experiments on the Kodak and CLIC 2024 datasets demonstrate that our QMF compression method consistently outperforms JPEG at low bit rates below 0.25 bits per pixel (bpp) and remains comparable at higher bit rates. We also assessed our method's capability to preserve visual semantics by evaluating an ImageNet pre-trained classifier on compressed images. Remarkably, our method improved top-1 accuracy by over 5 percentage points compared to JPEG at bit rates under 0.25 bpp. The project is available at https://github.com/pashtari/lrf .

eess.IV↗

arXiv2025

Forward-backward splitting under the light of generalized convexity

In this paper we present a unifying framework for continuous optimization methods grounded in the concept of generalized convexity. Utilizing the powerful theory of $Φ$-convexity, we propose a conceptual algorithm that extends the classical difference-of-convex method, encompassing a broad spectrum of optimization algorithms. Relying exclusively on the tools of generalized convexity we develop a gap function analysis that strictly characterizes the decrease of the function values, leading to simplified and unified convergence results. As an outcome of this analysis, we naturally obtain a generalized PL inequality which ensures $q$-linear convergence rates of the proposed method, incorporating various well-established conditions from the existing literature. Moreover we propose a $Φ$-Bregman proximal point interpretation of the scheme that allows us to capture conditions that lead to sublinear rates under convexity.

math.OC↗

arXiv2025

Blocked Cholesky factorization updates of the Riccati recursion using hyperbolic Householder transformations

Newton systems in quadratic programming (QP) methods are often solved using direct Cholesky or LDL factorizations. When the linear systems in successive iterations differ by a low-rank modification (as is common in active set and augmented Lagrangian methods), updating the existing factorization can offer significant performance improvements over recomputing a full Cholesky factorization. We review the hyperbolic Householder transformation, and demonstrate its usefulness in describing low-rank Cholesky factorization updates. By applying this hyperbolic Householder-based framework to the well-known Riccati recursion for solving saddle-point problems with optimal control structure, we develop a novel algorithm for updating the factorizations used in optimization solvers for optimal control. Specifically, the proposed method can be used to efficiently solve the semismooth Newton systems that are at the core of the augmented Lagrangian-based QPALM-OCP solver. An optimized open-source implementation of the proposed factorization update routines is provided as well.

math.OC↗

arXiv2025

Convergence of the Chambolle-Pock Algorithm in the Absence of Monotonicity

The Chambolle-Pock algorithm (CPA), also known as the primal-dual hybrid gradient method, has gained popularity over the last decade due to its success in solving large-scale convex structured problems. This work extends its convergence analysis for problems with varying degrees of (non)monotonicity, quantified through a so-called oblique weak Minty condition on the associated primal-dual operator. Our results reveal novel stepsize and relaxation parameter ranges which do not only depend on the norm of the linear mapping, but also on its other singular values. In particular, in nonmonotone settings, in addition to the classical stepsize conditions, extra bounds on the stepsizes and relaxation parameters are required. On the other hand, in the strongly monotone setting, the relaxation parameter is allowed to exceed the classical upper bound of two. Moreover, we build upon the recently introduced class of semimonotone operators, providing sufficient convergence conditions for CPA when the individual operators are semimonotone. Since this class of operators encompasses traditional operator classes including (hypo)- and co(hypo)-monotone operators, this analysis recovers and extends existing results for CPA. Tightness of the proposed stepsize ranges is demonstrated through several examples.

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