Absence Of Convexity Research Articles

Optimization Landscape of Policy Gradient Methods for Discrete-Time Static Output Feedback.

In recent times, significant advancements have been made in delving into the optimization landscape of policy gradient methods for achieving optimal control in linear time-invariant (LTI) systems. Compared with state-feedback control, output-feedback control is more prevalent since the underlying state of the system may not be fully observed in many practical settings. This article analyzes the optimization landscape inherent to policy gradient methods when applied to static output feedback (SOF) control in discrete-time LTI systems subject to quadratic cost. We begin by establishing crucial properties of the SOF cost, encompassing coercivity, L -smoothness, and M -Lipschitz continuous Hessian. Despite the absence of convexity, we leverage these properties to derive novel findings regarding convergence (and nearly dimension-free rate) to stationary points for three policy gradient methods, including the vanilla policy gradient method, the natural policy gradient method, and the Gauss-Newton method. Moreover, we provide proof that the vanilla policy gradient method exhibits linear convergence toward local minima when initialized near such minima. This article concludes by presenting numerical examples that validate our theoretical findings. These results not only characterize the performance of gradient descent for optimizing the SOF problem but also provide insights into the effectiveness of general policy gradient methods within the realm of reinforcement learning.

Decarbonization with induced technical change: Exploring the niche potential of hydrogen in heavy transportation

Fuel cells and electric batteries are competing technologies for the energy transition in heavy transportation. We explore the conditions for the survival of a unique technology in the long term. Learning by doing suggests focusing on a single technology while differentiation and decreasing return to scale (cost convexity) favor diversification. Exogenous technical change also plays a role. The interaction between these factors is analyzed in a general model. It is proved that in absence of convexity and exogenous technical change, only one technology is used for the whole transition. We then apply this framework to analyze the competition between fuel-cell electric buses (FCEBs) and battery electric buses (BEB) in the European bus sector, considering both learning by doing and exogenous technical change. The model is calibrated and solved. It is shown that the existence of a niche for FCEBs critically depends on the speed at which cost reductions are achieved. The speed depends both on the size of the niche and the rate of learning by doing for FCEBs. Public policies to decentralize the socially optimal trajectory in terms of carbon tax and subsidies to learning by doing are derived.

A New Existence Result of Equilibria for Vector Equilibrium Problems

In this paper, we deal with vector equilibrium problems. We prove the nonemptiness of the solution set for this type of problems in the sequential compactness case and in the absence of convexity and lower semicontinuity assumptions. Some examples are presented and an existence result for countable systems of vector equilibrium problems is stated.

A Sharp Analysis of Covariate Adjusted Precision Matrix Estimation via Alternating Projected Gradient Descent

In this letter, we present a sharp algorithmic analysis for alternating projected gradient descent which is used to solve the covariate adjusted precision matrix estimation problem in high-dimensional settings. By introducing a new analytical tool (the generic chaining), we remove the impractical resampling assumption used in the literature. The new analysis also demonstrates that this algorithm not only enjoys a linear convergence rate in the absence of convexity, but also attains the minimax rate with optimal order of sample complexity. Our results, meanwhile, reveal a time-data tradeoff in this problem. Numerical experiments are provided to verify our theoretical results.

Reduction method for functional nonconvex differential inclusions

Our aim in this paper is to present a reduction method that solves first order functional differential inclusion in the nonconvex case. This approach is based on a discretization of the time interval, a construction of approximate solutions by reducing the problem to a problem without delay and an application of known results in this case. We generalises earlier results, the right hand side of the inclusion has nonconvex values and satisfies a linear growth condition instead to be integrably bounded. The lack of convexity is replaced by the topological properties of decomposable sets, that represents a good alternative in the absence of convexity.

Higher order Langevin Monte Carlo algorithm

A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $\pi$ that has a density $\hat{\pi}$ on $\mathbb{R}^d$ known up to a normalizing constant. Moreover, $-\log \hat{\pi}$ is assumed to have a locally Lipschitz gradient and its third derivative is locally H\"{o}lder continuous with exponent $\beta \in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+ \beta/2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation even in the absence of convexity. Finally, in the case where $-\log \hat{\pi}$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.

Lyapunov stability and generalized invariance principle for nonconvex differential inclusions

This paper studies the system stability problems of a class of nonconvex differential inclusions. At first, a basic stability result is obtained by virtue of locally Lipschitz continuous Lyapunov functions. Moreover, a generalized invariance principle and related attraction conditions are proposed and proved to overcome the technical difficulties due to the absence of convexity. In the technical analysis, a novel set-valued derivative is proposed to deal with nonsmooth systems and nonsmooth Lyapunov functions. Additionally, the obtained results are consistent with the existing ones in the case of convex differential inclusions with regular Lyapunov functions. Finally, illustrative examples are given to show the effectiveness of the methods.

Data envelopment analysis in the absence of convexity: Specifying efficiency status and estimating returns to scale

The first purpose of this study is to present new DEA (data envelopment analysis) approaches that distinguish the efficiency status (efficient or inefficient) of decision-making units (DMUs) and estimate right- and left-hand returns to scale (RTS) of efficient DMUs in free disposal hull (FDH) models. The second one is to estimate most productive scale size (MPSS) in FDH models. The proposed RTS method has three preferences with respect to the previous RTS methods in this context. First one is that it is capable of identifying all FDH-efficient DMUs, whereas the previous methods may not be able to specify all FDH-efficient DMUs. Besides, the second one is to estimate right- and left-hand RTS of FDH-efficient DMUs, while the former methods are only capable of estimating RTS of FDH-efficient DMUs. Moreover, the third one is that it is always able to estimate right- and left-hand RTS, correctly, whereas the previous methods are sometimes capable of estimating RTS, correctly. These issues are the motivations for creating this current research. Finally, a numerical example and an empirical application are exhibited for illustrating and comparing the proposed method with the former methods. Then, some consequences are provided based upon the obtained results and also some suggestions for future works are presented. Proofs of some theorems and lemmas are given in the Appendix.

Topological fixed point theory and applications to variational inequalities

This is the first part of a work on generalized variational inequalities and their applications in optimization. It proposes a general theoretical framework for the solvability of variational inequalities with possibly non-convex constraints and objectives. The framework consists of a generic constrained nonlinear inequality ( $\exists\hat{u}\in\Psi(\hat {u})$ , $\exists \hat{y}\in\Phi(\hat{u})$ , with $\varphi(\hat{u},\hat{y},\hat{u})\leq \varphi(\hat{u},\hat{y},v)$ , $\forall v\in\Psi(\hat{u})$ ) derived from new topological fixed point theorems for set-valued maps in the absence of convexity. Simple homotopical and approximation methods are used to extend the Kakutani fixed point theorem to upper semicontinuous compact approachable set-valued maps defined on a large class of non-convex spaces having non-trivial Euler-Poincare characteristic and modeled on locally finite polyhedra. The constrained nonlinear inequality provides an umbrella unifying and extending a number of known results and approaches in the theory of generalized variational inequalities. Various applications to optimization problems will be presented in the second part to this work to be published ulteriorly.

An Inertial Tseng’s Type Proximal Algorithm for Nonsmooth and Nonconvex Optimization Problems

We investigate the convergence of a forward---backward---forward proximal-type algorithm with inertial and memory effects when minimizing the sum of a nonsmooth function with a smooth one in the absence of convexity. The convergence is obtained provided an appropriate regularization of the objective satisfies the Kurdyka---źojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions.

Cadzow Denoising Upgraded: A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements

We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpass-filtered samples. We show that maximum-likelihood estimation of the unknown parameters amounts to a difficult, even believed NP-hard, matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, based on a recently proposed splitting method for convex nonsmooth optimization. Although the algorithm comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same ease of implementation and speed.

A new approach to optimal control of ill-posed semilinear elliptic equations in the absence of convexity

An optimal control problem governed by a class of ill-posed semilinear elliptic equations in the absence of convexity is studied in this paper. Firstly, by using a variant of the penalization method, the Pontryagin’s maximum principle for the initial problem is obtained. Secondly, an existence result is also proved.

On nonlinear inclusions in non-smooth domains

This paper reviews the old and new landmark extensions of the famous intermediate value theorem (IVT) of Bolzano and Poincare to a set-valued operator \({\Phi : E \supset X \rightrightarrows E}\) defined on a possibly non- convex, non-smooth, or even non-Lipschitzian domain X in a normed space E. Such theorems are most general solvability results for nonlinear inclusions: \({\exists x_{0} \in X}\) with \({0 \in \Phi (x_{0}).}\) Naturally, the operator Φ must have continuity properties (essentially upper semi- or hemi-continuity) and its values (assumed to be non-empty closed sets) may be convex or have topological properties that extend convexity. Moreover, as the one-dimensional IVT simplest formulation tells freshmen calculus students, to have a zero, the mapping must also satisfy “direction conditions” on the boundary ∂X which, when \({X = [a,b] \subset E = \mathbb{R}}\), Φ (x) = f(x) is an ordinary single-valued continuous mapping, consist of the traditional “sign condition” f (a) f (b) ≤ 0. When X is a convex subset of a normed space, this sign condition is expressed in terms of a tangency boundary condition \({\Phi (x) \cap T_{X}(x) \neq \emptyset}\), where TX(x) is the tangent cone of convex analysis to X at \({x \in \partial X}\). Naturally, in the absence of convexity or smoothness of the domain X, the tangency condition requires the consideration of suitable local approximation concepts of non-smooth analysis, which will be discussed in the paper in relationship to the solvability of general dynamical systems. Open image in new window

Cutting Plane Oracles to Minimize Non-smooth Non-convex Functions

We discuss a bundle method for non-smooth non-convex optimization programs. In the absence of convexity, a substitute for the cutting plane mechanism has to be found. We propose such a mechanism and prove convergence of our method in the sense that every accumulation point of the sequence of serious iterates is critical.

Optimal control of multisolution ordinary differential equations in the absence of convexity

In this paper, the authors study an optimal control problem governed by a class of multistate ordinary differential equations in the absence of convexity. To overcome the difficulty that the corresponding approximate optimal control problem may have no solution, relaxed controls are introduced. With the help of relaxation theory, Pontryagin’s maximum principle for the optimal pairs of the original control problem is obtained. In the end of this paper, the authors discuss the application of the maximum principle by an example.

Convexity and Concavity Detection in Computational Graphs: Tree Walks for Convexity Assessment

We examine symbolic tools associated with two modeling systems for mathematical programming, which can be used to automatically detect the presence or absence of convexity and concavity in the objective and constraint functions, as well as convexity of the feasible set in some cases. The coconut solver system [Schichl, H. 2004a. COCONUT: COntinuous CONstraints—Updating the technology] focuses on nonlinear global continuous optimization and possesses its own modeling language and data structures. The Dr. Ampl meta-solver [Fourer, R., D. Orban. 2007. Dr. Ampl—A meta solver for optimization. Technical Report G-2007-10, GERAD, Montréal] aims to analyze nonlinear differentiable optimization models and hooks into the ampl Solver Library [Gay, D. M. 2002. Hooking your solver to AMPL]. Our symbolic convexity analysis may be supplemented, when it returns inconclusive results, with a numerical phase that may detect nonconvexity. We report numerical results using these tools on sets of test problems for both global and local optimization.

Functional Inequalities in the Absence of Convexity and Lower Semicontinuity with Applications to Optimization

In this paper we extend some results in [Dinh, Goberna, López, and Volle, Set-Valued Var. Anal., to appear] to the setting of functional inequalities when the standard assumptions of convexity and lower semicontinuity of the involved mappings are absent. This extension is achieved under certain condition relative to the second conjugate of the involved functions. The main result of this paper, Theorem 1, is applied to derive some subdifferential calculus rules and different generalizations of the Farkas lemma for nonconvex systems, as well as some optimality conditions and duality theory for infinite nonconvex optimization problems. Several examples are given to illustrate the significance of the main results and also to point out the potential of their applications to get various extensions of Farkas-type results and to the study of other classes of problems such as variational inequalities and equilibrium models.

Financial transmission rights in convex pool markets

This paper studies financial transmission rights in electricity pool markets with nodal pricing. We prove that simultaneous feasibility entails revenue adequacy in a general framework of convex optimization, and show by counterexample as to how this result might fail in the absence of convexity.

Globally convergent block-coordinate techniques for unconstrained optimization

In this paper we define new classes of globally convergent block-coordinate techniques for the unconstrained minimization of a continuously differentiable function. More specifically, we first describe conceptual models of decomposition algorithms based on the interconnection of elementary operations performed on the block components of the variable vector. Then we characterize the elementary operations defined through a suitable line search or the global minimization in a component subspace. Using these models, we establish new results on the convergence of the nonlinear Gauss–Seidel method and we prove that this method with a two-block decomposition is globally convergent towards stationary points, even in the absence of convexity or uniqueness assumptions. In the general case of nonconvex objective function and arbitrary decomposition we define new globally convergent line-search-based schemes that may also include partial global inimizations with respect to some component. Computational aspects are discussed and, in particular, an application to a learning problem in a Radial Basis Function neural network is illustrated.

New Existence Results for Optimal Controls in the Absence of Convexity: The Importance of Extremality

A new approach to “existence without convexity” is presented. Instead of applying a Weierstrass–Tonelli-type existence result to a convexified (i.e., relaxed) form of the control problem, which is standard, it uses a new extremum principle. This principle guarantees the existence of an optimal relaxed control function that is a convex combination of relaxation of ordinary control functions. The same linear relationship is maintained for the corresponding trajectories, and this is of much benefit in the final phase, which consists of a deconvexification by a global subgradient argument and Lyapunov’s theorem. The extremum principle is based on extreme point features of the relaxed control problem. It can be stated in an abstract form and extends the classical extremum principle of Bauer. The new approach applies both to optimal control problems of suitable type (dynamics linear in the state variable) and to abstract variational problems without dynamics, well-known in economics. With a different interpretation, it can also give bang-bang existence results. Several new applications serve to illustrate the power of the approach.