Nonsmooth Formulation Research Articles

The Boussinesq System with Nonsmooth Boundary Conditions: Existence, Relaxation, and Topology Optimization

In this paper, we tackle a topology optimization problem which consists in finding the optimal shape of a solid located inside a fluid that minimizes a given cost function. The motion of the fluid is modeled thanks to the Boussinesq system which involves the unsteady Navier-Stokes equation coupled to a heat equation. In order to cover several models presented in the literature, we choose a non-smooth formulation for the outlet boundary conditions and an optimization parameter of bounded variations. This paper aims at proving existence of solutions to the resulting equations, along with the study of a relaxation scheme of the non-smooth conditions. A second part covers the topology optimization problem itself for which we proved the existence of optimal solutions and provides the definition of first order necessary optimality conditions.

Low-Rank Matrix Recovery with Composite Optimization: Good Conditioning and Rapid Convergence

The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, nonsmooth penalty formulations do not suffer from the same type of ill-conditioning. Consequently, standard algorithms for nonsmooth optimization, such as subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within constant relative error of the solution. Moreover, nonsmooth formulations are naturally robust against outliers. Our framework subsumes such important computational tasks as phase retrieval, blind deconvolution, quadratic sensing, matrix completion, and robust PCA. Numerical experiments on these problems illustrate the benefits of the proposed approach.

Composite optimization for robust rank one bilinear sensing

Abstract We consider the task of recovering a pair of vectors from a set of rank one bilinear measurements, possibly corrupted by noise. Most notably, the problem of robust blind deconvolution can be modeled in this way. We consider a natural nonsmooth formulation of the rank one bilinear sensing problem and show that its moduli of weak convexity, sharpness and Lipschitz continuity are all dimension independent, under favorable statistical assumptions. This phenomenon persists even when up to half of the measurements are corrupted by noise. Consequently, standard algorithms, such as the subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within a constant relative error of the solution. We complete the paper with a new initialization strategy, complementing the local search algorithms. The initialization procedure is both provably efficient and robust to outlying measurements. Numerical experiments, on both simulated and real data, illustrate the developed theory and methods.

The nonsmooth landscape of phase retrieval

Abstract We consider a popular nonsmooth formulation of the real phase retrieval problem. We show that under standard statistical assumptions a simple subgradient method converges linearly when initialized within a constant relative distance of an optimal solution. Seeking to understand the distribution of the stationary points of the problem, we complete the paper by proving that as the number of Gaussian measurements increases, the stationary points converge to a codimension two set, at a controlled rate. Experiments on image recovery problems illustrate the developed algorithm and theory.

Nonsmooth Formulation for Handling Unclassified Process Streams in the Optimization of Work and Heat Exchange Networks

Pinch analysis provides a systematic methodology for improving efficiency through enhanced process integration. Originally, the methodology focused on heat integration with heat exchanger network (...

Fused Group Lasso Regularized Multi-Task Feature Learning and Its Application to the Cognitive Performance Prediction of Alzheimer's Disease.

Alzheimer's disease (AD) is characterized by gradual neurodegeneration and loss of brain function, especially for memory during early stages. Regression analysis has been widely applied to AD research to relate clinical and biomarker data such as predicting cognitive outcomes from MRI measures. Recently, multi-task based feature learning (MTFL) methods with sparsity-inducing [Formula: see text]-norm have been widely studied to select a discriminative feature subset from MRI features by incorporating inherent correlations among multiple clinical cognitive measures. However, existing MTFL assumes the correlation among all tasks is uniform, and the task relatedness is modeled by encouraging a common subset of features via sparsity-inducing regularizations that neglect the inherent structure of tasks and MRI features. To address this issue, we proposed a fused group lasso regularization to model the underlying structures, involving 1) a graph structure within tasks and 2) a group structure among the image features. To this end, we present a multi-task feature learning framework with a mixed norm of fused group lasso and [Formula: see text]-norm to model these more flexible structures. For optimization, we employed the alternating direction method of multipliers (ADMM) to efficiently solve the proposed non-smooth formulation. We evaluated the performance of the proposed method using the Alzheimer's Disease Neuroimaging Initiative (ADNI) datasets. The experimental results demonstrate that incorporating the two prior structures with fused group lasso norm into the multi-task feature learning can improve prediction performance over several competing methods, with estimated correlations of cognitive functions and identification of cognition-relevant imaging markers that are clinically and biologically meaningful.

Joint multicast beamforming and user scheduling in large‐scale antenna systems

This paper studies the joint multicast beamforming and user scheduling problem, with the objective of minimizing total transmitting power across multiple channels by jointly assigning each user to appropriate channel and designing multicast beamformer for each channel. The problem of interest is formulated in two different optimization problems, a mixed binary quadratically constrained quadratic program and a highly-structured nonsmooth program. Two different algorithms, based on convex relaxation and convex restriction, respectively, are proposed to solve the problem. The performance ratio between the approximate solution provided by the convex-relaxation-based algorithm and optimal solution is proved to be upper bounded by a constant independent of problem data. The convex-restriction-based algorithm is guaranteed to converge to a critical point to the nonsmooth formulation problem. Finally, extensive simulation results verify the theoretical analysis and demonstrate the advantage of the proposed co-design scheme over conventional fixed scheduling and random scheduling in terms of power consumption.

Linearly Convergent First-Order Algorithms for Semidefinite Programming

In this paper, we consider two formulations for Linear Matrix Inequalities (LMIs) under Slater type constraint qualification assumption, namely, SDP smooth and non-smooth formulations. We also propose two first-order linearly convergent algorithms for solving these formulations. Moreover, we introduce a bundle-level method which converges linearly uniformly for both smooth and non-smooth problems and does not require any smoothness information. The convergence properties of these algorithms are also discussed. Finally, we consider a special case of LMIs, linear system of inequalities, and show that a linearly convergent algorithm can be obtained under a weaker assumption.

Particle-based solid for nonsmooth multidomain dynamics

A method for simulation of elastoplastic solids in multibody systems with nonsmooth and multidomain dynamics is developed. The solid is discretised into pseudo-particles using the meshfree moving least squares method for computing the strain tensor. The particle’s strain and stress tensor variables are mapped to a compliant deformation constraint. The discretised solid model thus fit a unified framework for nonsmooth multidomain dynamics simulations including rigid multibodies with complex kinematic constraints such as articulation joints, unilateral contacts with dry friction, drivelines, and hydraulics. The nonsmooth formulation allows for impact impulses to propagate instantly between the rigid multibody and the solid. Plasticity is introduced through an associative perfectly plastic modified Drucker–Prager model. The elastic and plastic dynamics are verified for simple test systems, and the capability of simulating tracked terrain vehicles driving on a deformable terrain is demonstrated.

Testing Different Nonsmooth Formulations of the Lennard–Jones Potential in Atomic Clustering Problems

A cluster is a group of identical molecules or atoms loosely bound by inter-atomic forces. The optimal geometry minimises the potential energy--usually modelled as the Lennard---Jones potential--of the cluster. The minimisation of the Lennard---Jones potential is a very difficult global optimisation problem with extremely many local minima. In addition to cluster problems, the Lennard---Jones potential represents an important component in many of the potential energy models used, for example, in protein folding, protein---peptide docking, and complex molecular conformation problems. In this paper, we study different modifications of the Lennard---Jones potential in order to improve the success rate of finding the global minimum of the original potential. The main interest of the paper is in nonsmooth penalised form of the Lennard---Jones potential. The preliminary numerical experiments confirm that the success rate of finding the global minimum is clearly improved when using the new formulae.

A comparative study between the augmented Lagrangian method and the complementarity approach for modeling the contact problem

The increasing demand for real-time high-fidelity multibody dynamics simulations in several modern fields such as robotics and computer game industries has motivated many researches to propose novel approaches to model multibody systems with several contacts. The possibility of different contact conditions in a system with several contacts yields a combinatorial problem of potentially large size. Rigid contact model which is the most common model used for real-time simulations yields a non-smooth dynamic formulation. The solution of such a system can be governed using different methods. In this paper a comparison between the complementarity approaches and the augmented Lagrangian based formulations to deal with non-smooth contact models is presented via numerical examples, and the advantages and shortcomings of each method are discussed.

A Constructive Approach to Estimating Pure Characteristics Demand Models with Pricing

Discrete-choice demand models are important and fundamental tools for understanding consumers’ choice behavior and for analyzing firms’ operations and pricing strategies. In these models, products are often described as a vector of observed characteristics. A consumer chooses the product that maximizes her utility, assumed to be a function of the observed product characteristics and the consumer’s preference over these product characteristics. One central task in the demand estimation literature is to infer, based on observed data, consumers’ preferences on product characteristics. We consider such an estimation problem for pure characteristics models, a class of random coefficients demand models without the idiosyncratic logit error term in a consumer’s utility function. The absence of the logit error term and the use of numerical integration to approximate the integral in aggregate market shares lead to a nonsmooth formulation of approximated market share equations. As a result, solving the approximated market share equations and estimating the model by using existing methods proposed in the econometrics literature remain computationally intractable. To overcome this difficulty, we first characterize consumers’ purchase decisions by a system of complementarity constraints. This new characterization leads to smooth approximated market share equations and allows us to cast the corresponding generalized method of moments (GMM) estimation problem essentially as a quadratic program with linear complementarity constraints, parameterized by an exponential, thus nonlinear, function of the structural parameter on price. We also extend this estimation framework to incorporate an endogenous pricing mechanism that captures the competitive profit maximization behavior of the producing firms. We provide existence results of a solution for the GMM estimator and present numerical results to demonstrate the computational effectiveness of our approach.

Flying elephants: a general method for solving non-differentiable problems

Flying Elephants (FE) is a generalization and a new interpretation of the Hyperbolic Smoothing approach. The article introduces the fundamental smoothing procedures. It contains a general overview of successful applications of the approach for solving a select set of five important problems, namely: distance geometry, covering, clustering, Fermat---Weber and hub location. For each problem the original non-smooth formulation and the succedaneous completely differentiable one are presented. Computational experiments for all related problems obtained results that exhibited a high level of performance according to all criteria: consistency, robustness and efficiency. For each problem some results to illustrate the performance of FE are also presented.

Micro-rheology of dense particulate flows: Application to immersed avalanches

Abstract We rely here on a non-smooth contact dynamics (NSCD) approach to treat particle collisions in a direct numerical simulation of a dense particulate flow. Interactions between particles are considered by a non-smooth formulation of particle dynamics at the microscopic scale, which enables one to straightforwardly implement complex contact laws. The hydrodynamic coupling is achieved by a distributed Lagrange multiplier/fictitious domain (DLM/FD) method. As a preliminary step, the relevance of our NSCD-DLM/FD method is assessed by comparing results of 2D sedimentation simulations with those obtained with a usual molecular dynamics collision model. Then, we use it to investigate how a fully immersed granular packing collapses depending on its initial particle volume fraction, providing clues on the micro-rheology of dense particulate flows.

Exact Penalization of Mathematical Programs with Equilibrium Constraints

We study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints (MPECs). In the first part, we prove that a Mangasarian--Fromovitz-type condition ensures the existence of a stable local error bound at the root of a real-valued nonnegative piecewise smooth function. A specification to nonsmooth formulations of equilibrium constraints, e.g., complementarity conditions or normal equations, provides conditions which guarantee the existence of a nonsmooth exact penalty function for MPECs. In the second part, we study a trust region minimization method for a class of composite nonsmooth functions which comprises exact penalty functions arising from MPECs. We prove a global convergence result for the general method and incorporate a penalty update rule. A further specification results in an SQP trust region method for MPECs based on an \(\ell_1\) penalty function.