Convexity Assumptions Research Articles

Overdetermined problems and constant mean curvature surfaces in cones

We consider a partially overdetermined problem in a sector-like domain \Omega in a cone \Sigma in \mathbb{R}^N , N\geq 2 , and prove a rigidity result of Serrin type by showing that the existence of a solution implies that \Omega is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces \Gamma with boundary which satisfy a 'gluing' condition with respect to the cone \Sigma . We prove that if either the cone is convex or the surface is a radial graph then \Gamma must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.

Passivity-based generalization of primal–dual dynamics for non-strictly convex cost functions

In this paper, we revisit primal–dual dynamics for convex optimization and present a generalization of the dynamics based on the concept of passivity. We hypothesize that supplying a stable zero to one of the integrators in the dynamics allows one to eliminate the assumption of strict convexity on the cost function. Then, we prove this hypothesis based on the passivity paradigm together with the invariance principle for Carathéodory systems. Additionally, we show that the presented algorithm is also a generalization of existing augmented Lagrangian-based primal–dual dynamics and discuss the benefit of the presented generalization in terms of noise reduction and convergence speed. Finally, the presented algorithm is demonstrated through simulation of lighting control in a building.

Probabilistic tree-based representation for solving minimum cost integer flow problems with nonlinear non-convex cost functions

The minimum cost flow problem (MCFP) is the most generic variation of the network flow problem which aims to transfer a commodity throughout the network to satisfy demands. The problem size (in terms of the number of nodes and arcs) and the shape of the cost function are the most critical factors when considering MCFPs. Existing mathematical programming techniques often assume the cost functions to be linear or convex. Unfortunately, the linearity and convexity assumptions are too restrictive for modelling many real-world scenarios. In addition, many real-world MCFPs are large-scale, with networks having a large number of nodes and arcs. In this paper, we propose a probabilistic tree-based genetic algorithm (PTbGA) for solving large-scale minimum cost integer flow problems with nonlinear non-convex cost functions. We first compare this probabilistic tree-based representation scheme with the priority-based representation scheme, which is the most commonly-used representation for solving MCFPs. We then compare the performance of PTbGA with that of the priority-based genetic algorithm (PrGA), and two state-of-the-art mathematical solvers on a set of MCFP instances. Our experimental results demonstrate the superiority and efficiency of PTbGA in dealing with large-sized MCFPs, as compared to the PrGA method and the mathematical solvers.

Nekhoroshev estimates for steep real-analytic elliptic equilibrium points

We prove that steep real-analytic elliptic equilibrium points are exponentially stable, generalizing results which were known only under a convexity assumption. This proves the general case of a conjecture of Nekhoroshev. This result is also an important step in our proof that generically, both in a topological and measure-theoretical sense, equilibrium points are super-exponentially stable.

A generalized Ekeland's variational principle for vector equilibria

In this paper, we establish an Ekeland-type variational principle for vector valued bifunctions defined on complete metric spaces with values in locally convex spaces ordered by closed convex cones. The main improvement consists in widening the class of bifunctions for which the variational principle holds. In order to prove this principle, a weak notion of continuity for vector valued functions is considered, and some of its properties are presented. We also furnish an existence result for vector equilibria in absence of convexity assumptions, passing through the existence of approximate solutions of an optimization problem.

A Conjugate Gradient Method Based on a Modified Secant Relation for Unconstrained Optimization

Based on the modified secant relation given by Zhang and Xu and making use of the Dai and Liao approach, Babaie-Kafaki et al. presented a conjugate gradient method to solve unconstrained optimization. In this paper, we made some modifications on the conjugate gradient parameter proposed by Babaie-Kafaki et al. and obtained some attractive results in theory and practice. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan-Moré performance profiles.

An SGD-based meta-learner with “growing” descent

The paper considers the problem of accelerating the convergence of stochastic gradient descent (SGD) in an automatic way. Previous research puts forward such algorithms as Adagrad, Adadelta, RMSprop, Adam and etc. to adapt both the updates and learning rates to the slope of a loss function. However, these adaptive methods do not share the same regret bound as the gradient descent method. Adagrad provably achieves the optimal regret bound on the assumption of convexity but accumulates the squared gradients in the denominator that dramatically shrinks the learning rate. This research is aimed at introducing a generalized logistic map directly into the SGD method in order to automatically set its parameters to the slope of the logistic loss function. The optimizer based on the population may be considered as a meta-learner that learns how to tune both the learning rate and gradient updates with respect to the rate of population growth. The present study yields the “growing” descent method and a series of computational experiments to point out the benefits of the proposed meta-learner.

The Brunn–Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators

We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in Rn satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) p-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope method by Alvarez-Lasry-Lions. We also deal with the existence and log-concavity of positive viscosity eigenfunctions.

On the extension of the Hager–Zhang conjugate gradient method for vector optimization

The extension of the Hager–Zhang (HZ) nonlinear conjugate gradient method for vector optimization is discussed in the present research. In the scalar minimization case, this method generates descent directions whenever, for example, the line search satisfies the standard Wolfe conditions. We first show that, in general, the direct extension of the HZ method for vector optimization does not yield descent (in the vector sense) even when an exact line search is employed. By using a sufficiently accurate line search, we then propose a self-adjusting HZ method which possesses the descent property. The proposed HZ method with suitable parameters reduces to the classical one in the scalar minimization case. Global convergence of the new scheme is proved without regular restarts and any convex assumption. Finally, numerical experiments illustrating the practical behavior of the approach are presented, and comparisons with the Hestenes–Stiefel conjugate gradient and the steepest descent methods are discussed.

Fast, Asymptotically Efficient, Recursive Estimation in a Riemannian Manifold

Stochastic optimisation in Riemannian manifolds, especially the Riemannian stochastic gradient method, has attracted much recent attention. The present work applies stochastic optimisation to the task of recursive estimation of a statistical parameter which belongs to a Riemannian manifold. Roughly, this task amounts to stochastic minimisation of a statistical divergence function. The following problem is considered: how to obtain fast, asymptotically efficient, recursive estimates, using a Riemannian stochastic optimisation algorithm with decreasing step sizes. In solving this problem, several original results are introduced. First, without any convexity assumptions on the divergence function, we proved that, with an adequate choice of step sizes, the algorithm computes recursive estimates which achieve a fast non-asymptotic rate of convergence. Second, the asymptotic normality of these recursive estimates is proved by employing a novel linearisation technique. Third, it is proved that, when the Fisher information metric is used to guide the algorithm, these recursive estimates achieve an optimal asymptotic rate of convergence, in the sense that they become asymptotically efficient. These results, while relatively familiar in the Euclidean context, are here formulated and proved for the first time in the Riemannian context. In addition, they are illustrated with a numerical application to the recursive estimation of elliptically contoured distributions.

Sectional symmetry of solutions of elliptic systems in cylindrical domains

In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in [ 6 , 9 , 16 , 18 ].

Accelerated proximal incremental algorithm schemes for non-strongly convex functions

There have been a number of recent advances in accelerated gradient and proximal schemes for optimization of convex finite sum problems. Defazio introduced a simple accelerated scheme for incremental stochastic proximal algorithms inspired by gradient based methods like SAGA. He was able to prove O(1/k) convergence for non-smooth function but only under the assumption of strong convexity of component terms. We introduce a slight modification of his scheme, called MP-SAGA for which we can prove O(1/k) convergence without strong convexity, but for smooth functions. Numerical results show that our method has better or comparable convergence to Defazio's scheme, even for non-strongly convex functions. As important special cases, we also derive an accelerated schemes for a multi–class formulation of SVM as well as clustering based on the SON regularization. Finally, we introduce a simplification of Point–SAGA, called SP–SAGA for problems such as SON with large number of variables and sparse relation between variables and objective terms.

Convergence of Stochastic Proximal Gradient Algorithm

We study the extension of the proximal gradient algorithm where only a stochastic gradient estimate is available and a relaxation step is allowed. We establish convergence rates for function values in the convex case, as well as almost sure convergence and convergence rates for the iterates under further convexity assumptions. Our analysis avoid averaging the iterates and error summability assumptions which might not be satisfied in applications, e.g. in machine learning. Our proofing technique extends classical ideas from the analysis of deterministic proximal gradient algorithms.

A new hybrid conjugate gradient method for large-scale unconstrained optimization problem with non-convex objective function

The paper is aimed to employ a modified secant equation in the framework of the hybrid conjugate gradient method based on Andrei’s approach to solve large-scale unconstrained optimization problems. The CG parameter in the mentioned hybrid CG method is a convex combination of CG parameters corresponding to the Hestenes–Stiefel and Dai–Yuan algorithms. The main feature of these hybrid methods is that the search direction is the Newton direction. The modified secant equation is derived by means of the fifth-order tensor model to improve the curvature information of the objective function. Also, to achieve convergence for general function, the revised version of the method based on the linear combination of the mentioned secant equation and Li and Fukushima’s modified secant equation is suggested. Under proper conditions, globally convergence properties of the new hybrid CG algorithm even without convexity assumption on the objective function is studied. Numerical experiments on a set of test problems of the CUTEr collection are done; they demonstrate the practical effectiveness of the proposed hybrid conjugate gradient algorithm.

An Exact Quantized Decentralized Gradient Descent Algorithm

We consider the problem of decentralized consensus optimization, where the sum of $n$ smooth and strongly convex functions are minimized over $n$ distributed agents that form a connected network. In particular, we consider the case that the communicated local decision variables among nodes are quantized in order to alleviate the communication bottleneck in distributed optimization. We propose the Quantized Decentralized Gradient Descent (QDGD) algorithm, in which nodes update their local decision variables by combining the quantized information received from their neighbors with their local information. We prove that under standard strong convexity and smoothness assumptions for the objective function, QDGD achieves a vanishing mean solution error under customary conditions for quantizers. To the best of our knowledge, this is the first algorithm that achieves vanishing consensus error in the presence of quantization noise. Moreover, we provide simulation results that show tight agreement between our derived theoretical convergence rate and the numerical results.

Saddle point criteria for multi-dimensional control optimisation problem involving first-order PDE constraints

In this paper, the multi-dimensional control optimisation problem involving first-order PDE constraints is considered. The absolute value penalty function method is used to solve the considered problem. Further, the saddle point of the Lagrange functional defined for the aforesaid control optimisation problem is discussed. Thereafter, the exactness property of the absolute value penalty function method is studied with the help of saddle-point theory. Then, the relationship between a minimiser of the penalised multi-dimensional control optimisation problem and a saddle point of the aforesaid control problem is established under the convexity assumption. Moreover, we have verified the theoretical results with applications.

Subdifferentials of Value Functions in Nonconvex Dynamic Programming for Nonstationary Stochastic Processes

The main goal of this paper is to apply the machinery of variational analysis and generalized differentiation to study infinite horizon stochastic dynamic programming (DP) with discrete time in the Banach space setting without convexity assumptions. Unlike to standard stochastic DP with stationary Markov processes, we investigate here stochastic DP in $L^p$ spaces to deal with nonstationary stochastic processes, which describe a more flexible learning procedure for the decision-maker. Our main concern is to calculate generalized subgradients of the corresponding value function and to derive necessary conditions for optimality in terms of the stochastic Euler inclusion under appropriate Lipschitzian assumptions. The usage of the subdifferential formula for integral functionals on $L^p$ spaces allows us, in particular, to find verifiable conditions to ensure smoothness of the value function without any convexity and/or interiority assumptions.

A Decentralized Proximal-Gradient Method With Network Independent Step-Sizes and Separated Convergence Rates

This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and proximal updates, respectively. The proposed algorithm is closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but has a few advantages. First of all, agents use uncoordinated step-sizes, and the stable upper bounds on step-sizes are independent of network topologies. The step-sizes depend on local objective functions, and they can be as large as those of the gradient descent. Secondly, for the special case without non-smooth terms, linear convergence can be achieved under the strong convexity assumption. The dependence of the convergence rate on the objective functions and the network are separated, and the convergence rate of the new algorithm is as good as one of the two convergence rates that match the typical rates for the general gradient descent and the consensus averaging. We provide numerical experiments to demonstrate the efficacy of the introduced algorithm and validate our theoretical discoveries.

A data envelopment analysis (DEA) model for building energy benchmarking

Purpose Building energy benchmarking is required for adopting an energy certification scheme, promoting energy efficiency and reducing energy consumption. It demonstrates the current level of energy consumption, the value of potential energy improvement and the prospects for additional savings. This paper aims to create a new data envelopment analysis (DEA) model that overcomes the limitations of existing models for building energy benchmarking. Design/methodology/approach Data preparation: the findings of the literature search and subject matter experts’ inputs are used to construct the DEA model. Particularly, it is ensured that the included variables would not violate the fundamental assumption of DEA modeling, DEA convexity axiom. New DEA formulation: controllable and non-controllable variables, e.g. weather conditions, are differentiated in the new formulation. A new approach is used to identify outliers to avoid skewing the efficiency scores for the rest of the buildings under consideration. Efficiency analysis: three distinct efficiencies are computed and analyzed in benchmarking building energy: overall, pure technical, and scale efficiency. Findings The proposed DEA approach is successfully applied to a data set provided by a utility management and energy services company that is active in the multifamily housing industry. Building characteristics and energy consumption of 124 multifamily properties in 15 different states in the USA are found in the data set. Buildings in this data set are benchmarked using the new DEA energy benchmarking formulation. Building energy benchmarking is also conducted in a time series manner showing how a particular building performs across the period of 12 months compared with its peers. Originality/value The proposed research contributes to the body of knowledge in building energy benchmarking through developing a new outlier detection method to mitigate the impact of super-efficient and super-inefficient buildings on skewing the efficiency scores of the other buildings; avoiding ratio variables in the DEA formulation to adhere to the convexity assumption that existing DEA methods do not follow; and distinguishing between controllable and non-controllable variables in the DEA formulation. This research contributes to the state of practice through providing a new energy benchmarking tool for facility managers and building owners that strive to relatively rank the energy-efficiency of their properties and identify low-performing properties as investment targets to enhance energy efficiency.

Lagrange Multiplier Rules for Weakly Minimal Solutions of Compact-Valued Set Optimization Problems

Optimality conditions are established in terms of Lagrange–Fritz–John multipliers as well as Lagrange–Kuhn–Tucker multipliers for set optimization problems (without any convexity assumption) by using new scalarization techniques. Additionally, we indicate how these results may be applied to some particular weak vector equilibrium problems.