arXiv subjects

Alberto De Marchi

Publications and source records attributed to Alberto De Marchi.

At least 19 recordsLinked to original sources

arXiv2026

Mirror descent algorithms with logarithmic barriers

This work derives convergence guarantees for mirror descent and proximal mirror descent algorithms when a logarithmic barrier is used as a distance-generating function. Standard approaches cannot be applied when the solution lies on the boundary, where the Bregman divergence blows up. We show that, in a specific setting, both methods enjoy an $O(\log k / k)$ rate, which is also tight. In addition, our contributions include: (i) a new technique for handling the blow-up; (ii) a resolution of a gap in the theory of relative smoothness; and (iii) a comparison of the proposed approach with interior-point methods.

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arXiv2026

Dynamic Proximal Point Method for Unconstrained Minimization

In this work, we present a novel dynamic proximal point algorithm for unconstrained optimization. The method generates a sequence of proximal subproblems, where the quadratic regularization term is weighted by a diagonal matrix that is updated adaptively at each iteration. Each subproblem is solved using an inner Newton's method combined with a line search, which provides a global convergence mechanism for the nonlinear solver. At the outer level, the algorithm updates the reference point and adjusts the regularization parameter based on the performance of the inner Newton solver. We derive the reduced linear system used to compute the Newton step, define the corresponding merit function, and discuss practical approaches for constructing the diagonal scaling matrix from derivative information. The paper also provides implementation-oriented pseudocode and stopping criteria that are consistent with the proposed method.

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arXiv2026

Elastically safeguarded augmented Lagrangian methods

We investigate, theoretically and numerically, a class of elastically safeguarded augmented Lagrangian methods for nonlinear optimization problems with inequality and equality constraints. Safeguarded augmented Lagrangian methods are known to exhibit substantially stronger global convergence guarantees than their non-safeguarded counterparts, making them attractive in practice. A persistent limitation, however, is that existing methods rely on a fixed safeguard, whose a priori selection can be difficult and inherently limits adaptivity, e.g., with respect to problem scaling. We propose an adaptive safeguarding mechanism that allows the safeguard to grow dynamically, overcoming these drawbacks while preserving the desirable global convergence properties of variants with a fixed safeguard. We further establish complexity bounds in terms of worst-case iteration counts. Numerical experiments comparing ALGENCAN, a state-of-the-art solver with rigid safeguard, against an elastically safeguarded variant thereof confirm that elastic safeguarding yields consistent practical benefits.

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arXiv2026

Resolvent Moreau identities without monotonicity: theory and applications to Gabay duality, Douglas--Rachford and ADMM

Duality is most often defined as a relationship between convex functions. If those functions are nonconvex, classical duality breaks down. Notwithstanding, we show that another kind of duality still exists, not between the functions themselves, but between the so-called resolvent operators used to solve associated problems. In fact, this duality-like relationship holds for any set-valued mapping, and is a generalization of the Moreau's identity. We use this duality to study existing operator schemes and to design new ones. In particular, we show that the duality-like relationship Daniel Gabay illuminated between the Douglas--Rachford splitting (DR) and the Alternating Direction Method of Multipliers (ADMM) extends to nonmonotone inclusion problems. We use this relationship to provide explicit counterexamples to the convergence of ADMM in several open cases, by studying the (easier to analyse) DR scheme. Motivated by our observations, we design a class of convergent resolvent homotopy schemes and use them to solve nonconvex-regularised least absolute deviations problems. This important problem class has received little attention in the literature, since the convex component of the objective does not enjoy strong convexity.

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arXiv2026

Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization

We introduce a proximal limited--memory quasi--Newton scheme for minimizing the sum of a continuously differentiable function and a proper, lower semicontinuous and prox-bounded, possibly nonsmooth, function. Both functions might be nonconvex. The method builds upon the computation of scaled proximal operators and is globalized by adaptively updating a regularization parameter based on a criterion of sufficient decrease. We prove global convergence under mild assumptions and then establish convergence of the entire sequence (with rates) under the Kurdyka--Lojasiewicz property. To efficiently solve the subproblems, we exploit the compact representation of limited-memory quasi-Newton updates. We derive also a compact representation of the limited--memory Kleinmichel formula, a rank-one quasi-Newton scheme that preserves positive definiteness under the same condition as the BFGS update. Numerical results show a significant speed up compared to other methods.

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arXiv2025

Augmented Lagrangian methods for fully convex composite optimization

This paper is concerned with augmented Lagrangian methods for the treatment of fully convex composite optimization problems. We extend the classical relationship between augmented Lagrangian methods and the proximal point algorithm to the inexact and safeguarded scheme in order to state global primal-dual convergence results. Our analysis distinguishes the regular case, where a stationary minimizer exists, and the irregular case, where all minimizers are nonstationary. Furthermore, we suggest an elastic modification of the standard safeguarding scheme which preserves primal convergence properties while guaranteeing convergence of the dual sequence to a multiplier in the regular situation. Although important for nonconvex problems, the standard safeguarding mechanism leads to weaker convergence guarantees for convex problems than the classical augmented Lagrangian method. Our elastic safeguarding scheme combines the advantages of both while avoiding their shortcomings.

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arXiv2025

Cutting plane methods with gradient-based heuristics

Cutting plane methods, particularly outer approximation, are a well-established approach for solving nonlinear discrete optimization problems without relaxing the integrality of decision variables. While powerful in theory, their computational performance can be highly variable. Recent research has shown that constructing cutting planes at the projection of infeasible points onto the feasible set can significantly improve the performance of cutting plane approaches. Motivated by this, we examine whether constructing cuts at feasible points closer to the optimal solution set could further enhance the effectiveness of cutting plane methods. We propose a hybrid method that combines the global convergence guarantees of cutting plane methods with the local exploration capabilities of first-order optimization techniques. Specifically, we use projected gradient methods as a heuristic to identify promising regions of the solution space and generate tighter, more informative cuts. We focus on binary optimization problems with convex differentiable objective functions, where projection operations can be efficiently computed via mixed-integer linear programming. By constructing cuts at points closer to the optimal solution set and eliminating non-optimal regions, the algorithm achieves better approximation of the feasible region and faster convergence. Numerical experiments confirm that our approach improves both the quality of the solution and computational efficiency across different solver configurations. This framework provides a flexible foundation for further extensions to more general discrete domains and offers a promising heuristic to the toolkit for nonlinear discrete optimization.

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arXiv2025

A condensing approach for linear-quadratic optimization with geometric constraints

Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and cardinality constraints, among others. In particular, we cover also situations where parts of the constraints are nonconvex and possibly complicated, but it is practical to compute projections onto this nonconvex set. Our approach combines the augmented Lagrangian framework with a solver-agnostic structure-exploiting subproblem reformulation. While convergence guarantees follow from the former, the proposed condensing technique leads to significant improvements in computational performance.

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arXiv2025

Reinforcement learning for adaptive interior point methods in convex quadratic programming

Quadratic programming is a workhorse of modern nonlinear optimization, control, and data science. Although regularized methods offer convergence guarantees under minimal assumptions on the problem data, they can exhibit the slow tail-convergence typical of first-order schemes, thus requiring many iterations to achieve high-accuracy solutions. Moreover, hyperparameter tuning significantly impacts the solver performance but how to find an appropriate parameter configuration remains an elusive research question. To address these issues, we explore how data-driven approaches can accelerate the solution process. Aiming at high-accuracy solutions, we focus on a regularized interior-point solver and carefully handle its two-loop flow and control parameters. We will show that reinforcement learning can make a significant contribution to facilitating the solver tuning and to speeding up the optimization process. Numerical experiments demonstrate that, after a lightweight training, the learned policy generalizes well to different problem classes with varying dimensions.

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arXiv2024

Affordable mixed-integer Lagrangian methods: optimality conditions and convergence analysis

Necessary optimality conditions in Lagrangian form and the sequential minimization framework are extended to mixed-integer nonlinear optimization, without any convexity assumptions. Building upon a recently developed notion of local optimality for problems with polyhedral and integrality constraints, a characterization of local minimizers and critical points is given for problems including also nonlinear constraints. This approach lays the foundations for developing affordable sequential minimization algorithms with convergence guarantees to critical points from arbitrary initializations. A primal-dual perspective, a local saddle point property, and the dual relationships with the proximal point algorithm are also advanced in the presence of integer variables. Preliminary numerical results are presented for an augmented Lagrangian and an interior point method.

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arXiv2024

A penalty barrier framework for nonconvex constrained optimization

We consider minimization problems with structured objective function and smooth constraints, and present a flexible framework that combines the beneficial regularization effects of (exact) penalty and interior-point methods. In the fully nonconvex setting, a pure barrier approach requires careful steps when approaching the infeasible set, thus hindering convergence. We show how a tight integration with a penalty scheme mitigates this issue and enables the construction of subproblems whose domain is independent of the explicit constraints. This decoupling allows us to leverage efficient solvers designed for unconstrained or suitably structured optimization tasks. The key behind all this is a marginalization step: closely related to a conjugacy operation, this step effectively merges (exact) penalty and barrier into a smooth, full domain functional object. When the penalty exactness takes effect, the generated subproblems do not suffer the ill-conditioning typical of barrier methods, nor do they exhibit the nonsmoothness of exact penalty terms. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. Stronger conclusions are available for the convex setting, where optimality can be guaranteed. Illustrative examples and numerical simulations demonstrate the wide range of problems our theory and algorithm are able to cover.

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arXiv2024

Hybrid optimal control with mixed-integer Lagrangian methods

Models involving hybrid systems are versatile in their application but difficult to optimize efficiently due to their combinatorial nature. This work presents a method to cope with hybrid optimal control problems which, in contrast to decomposition techniques, does not require relaxing the integrality constraints. Based on the discretize-then-optimize approach, our scheme addresses mixed-integer nonlinear problems under mild assumptions. The proposed numerical algorithm builds upon the augmented Lagrangian framework, whose subproblems are handled using successive mixed-integer linearizations with trust regions. We validate the performance of the numerical routine with extensive investigations using hybrid optimal control problems from different fields of application. Promising preliminary results are presented for a motion planning task with hysteresis and a Lotka-Volterra fishing problem with total variation.

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arXiv2023

Collision Avoidance using Iterative Dynamic and Nonlinear Programming with Adaptive Grid Refinements

Nonlinear optimal control problems for trajectory planning with obstacle avoidance present several challenges. While general-purpose optimizers and dynamic programming methods struggle when adopted separately, their combination enabled by a penalty approach is capable of handling highly nonlinear systems while overcoming the curse of dimensionality. Nevertheless, using dynamic programming with a fixed state space discretization limits the set of reachable solutions, hindering convergence or requiring enormous memory resources for uniformly spaced grids. In this work we solve this issue by incorporating an adaptive refinement of the state space grid, splitting cells where needed to better capture the problem structure while requiring less discretization points overall. Numerical results on a space manipulator demonstrate the improved robustness and efficiency of the combined method with respect to the single components.

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arXiv2023

Mixed-integer linearity in nonlinear optimization: a trust region approach

Bringing together nonlinear optimization with polyhedral and integrality constraints enables versatile modeling, but poses significant computational challenges. We investigate a method to address these problems based on sequential mixed-integer linearization with trust region safeguard, computing feasible iterates via calls to a generic mixed-integer linear solver. Convergence to critical, possibly suboptimal, feasible points is established for arbitrary starting points. Finally, we present numerical applications in nonsmooth optimal control and optimal network design and operation.

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arXiv2023

Local properties and augmented Lagrangians in fully nonconvex composite optimization

A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.

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arXiv2023

Implicit augmented Lagrangian and generalized optimization

Generalized nonlinear programming is considered without any convexity assumption, capturing a variety of problems that include nonsmooth objectives, combinatorial structures, and set-membership nonlinear constraints. We extend the augmented Lagrangian framework to this broad problem class, preserving an implicit formulation and introducing slack variables merely as a formal device. This, however, gives rise to a generalized augmented Lagrangian function that lacks regularity, due to the marginalization with respect to slack variables. Based on parametric optimization, we develop a tailored stationarity concept to better qualify the iterates, generated as approximate solutions to a sequence of subproblems. Using this variational characterization and the lifted representation, a suitable multiplier update rule is derived, and then asymptotic properties and convergence guarantees are established for a safeguarded augmented Lagrangian scheme. An illustrative numerical example showcases the modelling versatility gained by dropping convexity assumptions and indicates the practical benefits of the advocated implicit approach.

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arXiv2023

Relativistic Digital Twin: Bringing the IoT to the Future

Complex IoT ecosystems often require the usage of Digital Twins (DTs) of their physical assets in order to perform predictive analytics and simulate what-if scenarios. DTs are able to replicate IoT devices and adapt over time to their behavioral changes. However, DTs in IoT are typically tailored to a specific use case, without the possibility to seamlessly adapt to different scenarios. Further, the fragmentation of IoT poses additional challenges on how to deploy DTs in heterogeneous scenarios characterized by the usage of multiple data formats and IoT network protocols. In this paper, we propose the Relativistic Digital Twin (RDT) framework, through which we automatically generate general-purpose DTs of IoT entities and tune their behavioral models over time by constantly observing their real counterparts. The framework relies on the object representation via the Web of Things (WoT), to offer a standardized interface to each of the IoT devices as well as to their DTs. To this purpose, we extended the W3C WoT standard in order to encompass the concept of behavioral model and define it in the Thing Description (TD) through a new vocabulary. Finally, we evaluated the RDT framework over two disjoint use cases to assess its correctness and learning performance, i.e., the DT of a simulated smart home scenario with the capability of forecasting the indoor temperature, and the DT of a real-world drone with the capability of forecasting its trajectory in an outdoor scenario. Experiments show that the generated DT can estimate the behavior of its real counterpart after an observation stage, regardless of the considered scenario.

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arXiv2022

Proximal gradient methods beyond monotony

We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal gradient methods, which often rely on a line search procedure as globalization mechanism. We consider an adaptive nonmonotone proximal gradient scheme based on an averaged merit function and establish asymptotic convergence guarantees under weak assumptions, delivering results on par with the monotone strategy. Global worst-case rates for the iterates and a stationarity measure are also derived. Finally, a numerical example indicates the potential of nonmonotonicity and spectral approximations.

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