Nonconvex Function Research Articles

Generalized Derivatives and Optimality Conditions in Nonconvex Optimization

In this paper, we study the radial epiderivative notion for nonconvex functions, which extends the (classical) directional derivative concept. The paper presents new definition and new properties for this notion and establishes relationships between the radial epiderivative, the Clarke’s directional derivative, the Rockafellar’s subderivative and the directional derivative. The radial epiderivative notion is used to establish new regularity conditions without convexity conditions. The paper presents explicit formulations for computing the radial epiderivatives in terms of weak subgradients and vice versa. We also present an iterative algorithm for approximate computing of radial epiderivatives and show that the algorithm terminates in a finite number of iterations. The paper analyzes necessary and sufficient conditions for global optimums in nonconvex optimization via the radial epiderivatives. We formulate a necessary and sufficient condition for a global descent direction for radially epidifferentiable nonconvex functions. All the properties and theorems presented in this paper are illustrated and interpreted on examples.

Large-Scale Non-convex Stochastic Constrained Distributionally Robust Optimization

Distributionally robust optimization (DRO) is a powerful framework for training robust models against data distribution shifts. This paper focuses on constrained DRO, which has an explicit characterization of the robustness level. Existing studies on constrained DRO mostly focus on convex loss function, and exclude the practical and challenging case with non-convex loss function, e.g., neural network. This paper develops a stochastic algorithm and its performance analysis for non-convex constrained DRO. The computational complexity of our stochastic algorithm at each iteration is independent of the overall dataset size, and thus is suitable for large-scale applications. We focus on the general Cressie-Read family divergence defined uncertainty set which includes chi^2-divergences as a special case. We prove that our algorithm finds an epsilon-stationary point with an improved computational complexity than existing methods. Our method also applies to the smoothed conditional value at risk (CVaR) DRO.

Near-Optimal Resilient Aggregation Rules for Distributed Learning Using 1-Center and 1-Mean Clustering with Outliers

Byzantine machine learning has garnered considerable attention in light of the unpredictable faults that can occur in large-scale distributed learning systems. The key to secure resilience against Byzantine machines in distributed learning is resilient aggregation mechanisms. Although abundant resilient aggregation rules have been proposed, they are designed in ad-hoc manners, imposing extra barriers on comparing, analyzing, and improving the rules across performance criteria. This paper studies near-optimal aggregation rules using clustering in the presence of outliers. Our outlier-robust clustering approach utilizes geometric properties of the update vectors provided by workers. Our analysis show that constant approximations to the 1-center and 1-mean clustering problems with outliers provide near-optimal resilient aggregators for metric-based criteria, which have been proven to be crucial in the homogeneous and heterogeneous cases respectively. In addition, we discuss two contradicting types of attacks under which no single aggregation rule is guaranteed to improve upon the naive average. Based on the discussion, we propose a two-phase resilient aggregation framework. We run experiments for image classification using a non-convex loss function. The proposed algorithms outperform previously known aggregation rules by a large margin with both homogeneous and heterogeneous data distributions among non-faulty workers. Code and appendix are available at https://github.com/jerry907/AAAI24-RASHB.

A Bregman Proximal Stochastic Gradient Method with Extrapolation for Nonconvex Nonsmooth Problems

In this paper, we explore a specific optimization problem that involves the combination of a differentiable nonconvex function and a nondifferentiable function. The differentiable component lacks a global Lipschitz continuous gradient, posing challenges for optimization. To address this issue and accelerate the convergence, we propose a Bregman proximal stochastic gradient method with extrapolation (BPSGE), which only requires smooth adaptivity of the differentiable part. Under variance reduction framework, we not only analyze the subsequential and global convergence of the proposed algorithm under certain conditions, but also analyze the sublinear convergence rate of the subsequence, and the complexity of the algorithm, revealing that the BPSGE algorithm requires at most O(epsilon\^\,(-2)) iterations in expectation to attain an epsilon-stationary point. To validate the effectiveness of our proposed algorithm, we conduct numerical experiments on three real-world applications: graph regularized nonnegative matrix factorization (NMF), matrix factorization with weakly-convex regularization, and NMF with nonconvex sparsity constraints. These experiments demonstrate that BPSGE is faster than the baselines without extrapolation. The code is available at: https://github.com/nothing2wang/BPSGE-Algorithm.

Provably Convergent Federated Trilevel Learning

Trilevel learning, also called trilevel optimization (TLO), has been recognized as a powerful modelling tool for hierarchical decision process and widely applied in many machine learning applications, such as robust neural architecture search, hyperparameter optimization, and domain adaptation. Tackling TLO problems has presented a great challenge due to their nested decision-making structure. In addition, existing works on TLO face the following key challenges: 1) they all focus on the non-distributed setting, which may lead to privacy breach; 2) they do not offer any non-asymptotic convergence analysis which characterizes how fast an algorithm converges. To address the aforementioned challenges, this paper proposes an asynchronous federated trilevel optimization method to solve TLO problems. The proposed method utilizes u-cuts to construct a hyper-polyhedral approximation for the TLO problem and solve it in an asynchronous manner. We demonstrate that the proposed u-cuts are applicable to not only convex functions but also a wide range of non-convex functions that meet the u-weakly convex assumption. Furthermore, we theoretically analyze the non-asymptotic convergence rate for the proposed method by showing its iteration complexity to obtain ϵ-stationary point is upper bounded by O(1/ϵ²). Extensive experiments on real-world datasets have been conducted to elucidate the superiority of the proposed method, e.g., it has a faster convergence rate with a maximum acceleration of approximately 80%.

Sample-and-Bound for Non-convex Optimization

Standard approaches for global optimization of non-convex functions, such as branch-and-bound, maintain partition trees to systematically prune the domain. The tree size grows exponentially in the number of dimensions. We propose new sampling-based methods for non-convex optimization that adapts Monte Carlo Tree Search (MCTS) to improve efficiency. Instead of the standard use of visitation count in Upper Confidence Bounds, we utilize numerical overapproximations of the objective as an uncertainty metric, and also take into account of sampled estimates of first-order and second-order information. The Monte Carlo tree in our approach avoids the usual fixed combinatorial patterns in growing the tree, and aggressively zooms into the promising regions, while still balancing exploration and exploitation. We evaluate the proposed algorithms on high-dimensional non-convex optimization benchmarks against competitive baselines and analyze the effects of the hyper parameters.

GIPC: Fast and Stable Gauss-Newton Optimization of IPC Barrier Energy

Barrier functions are crucial for maintaining an intersection- and inversion-free simulation trajectory but existing methods, which directly use distance can restrict implementation design and performance. We present an approach to rewriting the barrier function for arriving at an efficient and robust approximation of its Hessian. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, from which analytic eigensystems are derived and enhanced with filtering and stiffening terms that ensure robustness with respect to the convergence of a Project-Newton solver. A further advantage of our rewriting of the barrier function is that it naturally caters to the notorious case of nearly parallel edge-edge contacts for which we also present a novel analytic eigensystem. Our approach is thus well suited for standard second-order unconstrained optimization strategies for resolving contacts, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The efficiency of our eigensystems alone yields a 3× speedup over the standard Incremental Potential Contact (IPC) barrier formulation. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.

Analysis of a Two-Step Gradient Method with Two Momentum Parameters for Strongly Convex Unconstrained Optimization

The paper is devoted to the theoretical and numerical analysis of the two-step method, constructed as a modification of Polyak’s heavy ball method with the inclusion of an additional momentum parameter. For the quadratic case, the convergence conditions are obtained with the use of the first Lyapunov method. For the non-quadratic case, sufficiently smooth strongly convex functions are obtained, and these conditions guarantee local convergence.An approach to finding optimal parameter values based on the solution of a constrained optimization problem is proposed. The effect of an additional parameter on the convergence rate is analyzed. With the use of an ordinary differential equation, equivalent to the method, the damping effect of this parameter on the oscillations, which is typical for the non-monotonic convergence of the heavy ball method, is demonstrated. In different numerical examples for non-quadratic convex and non-convex test functions and machine learning problems (regularized smoothed elastic net regression, logistic regression, and recurrent neural network training), the positive influence of an additional parameter value on the convergence process is demonstrated.

DF-ParPINN: parallel PINN based on velocity potential field division and single time slice focus

Modified Benney-Luke equation (mBL equation) is a three-dimensional temporal-spatial equation with complex structures, that is a high-dimensional partial differential equation (PDE), it is also a new equation of the physical ocean field, and its solution is important for studying the internal wave-wave interaction of inclined seafloor. For conventional PDE solvers such as the pseudo-spectral method, it is difficult to solve mBL equation with both accuracy and speed. Physics-informed neural network (PINN) incorporates physical prior knowledge in deep neural networks, which can solve PDE with relative accuracy and speed. However, PINN is only suitable for solving low-dimensional PDE with simple structures, and not suitable for solving high-dimensional PDE with complex structures. This is mainly because high-dimensional PDEs usually have complex structures and high-order derivatives and are likely to be high-dimensional non-convex functions, and the high-dimensional non-convex optimization problem is an NP-hard problem, resulting in the PINN easily falling into inaccurate local optimal solutions when solving high-dimensional PDEs. Therefore, we improve the PINN for the characteristics of mBL equation and propose “DF-ParPINN: parallel PINN based on velocity potential field division and single time slice focus” to solve mBL equation with large amounts of data. DF-ParPINN consists of three modules: temporal-spatial division module of overall velocity potential field, data rational selection module of multiple time slices, and parallel computation module of high-velocity fields and low-velocity fields. The experimental results show that the solution time of DF-ParPINN is no more than 0.5s, and its accuracy is much higher than that of PINN, PIRNN, cPINN, and DeepONet. Moreover, the relative error of DF-ParPINN after deep training 1000000 epochs can be reduced to less than 0.1. The validity of DF-ParPINN proves that the improved PINN also can solve high dimensional PDE with complex structures and large amounts of data quickly and accurately, which is of great significance to the deep learning of the physical ocean field.

Generalized Nash equilibrium problems with mixed-integer variables

AbstractWe consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido–Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

Total variation image reconstruction algorithm based on non-convex function

Total variation image reconstruction algorithm based on non-convex function

On generalized Nash equilibrium problems in infinite-dimensional spaces using Nikaido–Isoda type functionals

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE.

Decentralized gradient tracking with local steps

Gradient tracking (GT) is an algorithm designed for solving decentralized optimization problems over a network (such as training a machine learning model). A key feature of GT is a tracking mechanism that allows us to overcome data heterogeneity between nodes. We develop a novel decentralized tracking mechanism, K-GT, which enables communication-efficient local updates in GT while inheriting the data-independence property of GT. We prove a convergence rate for K-GT on smooth non-convex functions and prove that it reduces the communication overhead asymptotically by a linear factor K, where K denotes the number of local steps. We illustrate the robustness and effectiveness of this heterogeneity correction on convex and non-convex benchmark problems and a non-convex neural network training task with the MNIST dataset.

Smoothing algorithms for nonsmooth optimization over the Stiefel manifold with applications to the graph Fourier basis problem

In this paper, we consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with a linear mapping. Besides, we are interested in its application to the graph Fourier basis problem. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms.

Fabrication uncertainty guided design optimization of a photonic crystal cavity by using Gaussian processes

We present a fabrication uncertainty aware and robust design optimization approach that can be used to obtain robust design estimates for nonlinear, nonconvex, and expensive model functions. It is founded on Gaussian processes and a Monte Carlo sampling procedure, and assumes knowledge about the uncertainties associated with a manufacturing process. The approach itself is iterative. First, a large parameter domain is sampled in a coarse fashion. This coarse sampling is used primarily to determine smaller candidate regions to investigate in a second, more refined sampling pass. This finer step is used to obtain an estimate of the expected performance of the found design parameter under the assumed manufacturing uncertainties. We apply the presented approach to the robust optimization of the Purcell enhancement of a photonic crystal nanobeam cavity. We obtain a predicted robust Purcell enhancement of F¯P≈3.6. For comparison we also perform an optimization without robustness. We find that an unrobust optimum of FP≈256.5 dwindles to only F¯P≈0.7 when fabrication uncertainties are taken into account. We thus demonstrate that the presented approach is able to find designs of significantly higher performance than those obtained with conventional optimization.

A novel non-convex penalty function-based STAP algorithm for airborne passive radar systems

In conventional sparse Space-Time Adaptive Processing (STAP) algorithms, the l0-norm is often used instead of the l1-norm to relax convexity constraints. The approach presented in this paper can be mathematically reformulated as a linear problem, which can be efficiently solved using various convex optimization techniques. While this method effectively circumvents the NP-Hard complexity associated with the l0-norm, it does come with certain potential challenges. For example, the observation matrix must meet the criteria of the Restricted Isometry Property (RIP) and other stringent conditions. In this research, a novel STAP algorithm is introduced, which is based on a non-convex penalty function. This innovative algorithm substitutes the traditional l0-norm with a custom-designed non-convex penalty function and is applied to the Direct Filter Processor (DFP), solved using the recursive least squares (RLS) algorithm. The resulting algorithm, known as gp-RLS, is demonstrated through Monte Carlo simulations to outperform other l1-based and reduced-rank STAP algorithms in terms of faster convergence, improved signal-to-clutter-plus-noise ratio (SCNR), and enhanced target detection performance.

Feature extraction of small underwater sonar targets using neighborhood discriminant analysis

Linear discriminant analysis (LDA) stands as a widely used supervised feature extraction technique that maps data onto a low-dimensional subspace such that the between-class scatter is maximized while within-class scatter is minimized. Despite its utility, LDA faces many challenges, particularly when the within-class scatter matrix becomes singular due to small sample size. Other dimensionality reduction techniques, such as Neighbourhood Component Analysis (NCA), have been proposed as alternatives to LDA. NCA learns a linear transformation that maximizes the likelihood that datapoints of the same class are clustered together in the lower-dimensional space. However, the optimization of NCA relies on a non-convex cost function, making it prone to local minima. To address the challenges faced by both LDA and NCA, we propose a novel dimensionality reduction method named neighborhood discriminant analysis (NDA). Like NCA, NDA learns a linear transformation that aims to cluster datapoints based on class label. However, NDA is framed as an eigendecomposition problem, eliminating the need for non-convex optimization. We demonstrate the new approach on real small target sonar data. [Work funded by the ONR grant numbers N000142112420 and N000142312503, and DoD Navy (NEEC) Grant No. N001742010016.]

Application of the Parabola Method in Nonconvex Optimization

We consider the Golden Section and Parabola Methods for solving univariate optimization problems. For multivariate problems, we use these methods as line search procedures in combination with well-known zero-order methods such as the coordinate descent method, the Hooke and Jeeves method, and the Rosenbrock method. A comprehensive numerical comparison of the obtained versions of zero-order methods is given in the present work. The set of test problems includes nonconvex functions with a large number of local and global optimum points. Zero-order methods combined with the Parabola method demonstrate high performance and quite frequently find the global optimum even for large problems (up to 100 variables).

An Inertial Parametric Douglas–Rachford Splitting Method for Nonconvex Problems

In this paper, we propose an inertial parametric Douglas–Rachford splitting method for minimizing the sum of two nonconvex functions, which has a wide range of applications. The proposed algorithm combines the inertial technique, the parametric technique, and the Douglas–Rachford method. Subsequently, in theoretical analysis, we construct a new merit function and establish the convergence of the sequence generated by the inertial parametric Douglas–Rachford splitting method. Finally, we present some numerical results on nonconvex feasibility problems to illustrate the efficiency of the proposed method.

A comparison of alternative models for solving a non-linear single plant Hydro Unit Commitment problem

A wide range of real world optimization problems involves continuous decisions and non-linearities. Each non-linear component of such problems can be modeled either linearly or non-linearly, considering or not additional integer variables. This results into different modeling choices that can drastically impact the solution time and quality. In this paper, we evaluate representative modeling alternatives, including common models from the literature as well as new models featuring less common functions. The single plant Hydro Unit Commitment problem (1-HUC) is the considered non-linear use case. Among the non-linearities of the 1-HUC, we focus on those involved in the power production, more precisely the head effect and the turbines efficiency. The power is defined as a two-dimensional non-convex and non-concave function of the water flow and head decision variables, the latter being itself a one-dimensional concave function of the turbined volume. We consider both the general problem and a common special case, assuming that the water head is fixed. Several available solvers are used for each non-linear model and the best virtual solver is retained to focus on the model capabilities rather than on the solver performance. Based on the numerical experiments, three models stand out as the most efficient in terms of computational time, solution quality and feasibility, sometimes in a counter-intuitive manner. For each of these models, a solver is highlighted as the most adequate.