Showing papers on "Concave function published in 2014"

Generalized Nonconvex Nonsmooth Low-Rank Minimization

Abstract: As surrogate functions of 0-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex low-rank minimization problem is much more challenging than the nonconvex sparse minimization problem. We observe that all the existing nonconvex penalty functions are concave and monotonically increasing on [0, ∞). Thus their gradients are decreasing functions. Based on this property, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to solve the nonconvex nonsmooth low-rank minimization problem. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem. By setting the weight vector as the gradient of the concave penalty function, the WSVT problem has a closed form solution. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthetic data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.

Kantorovich duality for general transport costs and applications

Abstract: We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.

Arithmetic Geometry of Toric Varieties: Metrics, Measures and Heights

Abstract: We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real Monge-Amp\`ere measures, and Legendre-Fenchel duality. We also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This allows us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes. We also compute the height of toric projective curves with respect to the Fubini-Study metric, and of some toric bundles.

Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities

Abstract: We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prekopa–Leindler and Brascamp–Lieb inequalities. Further issues that we transpose to this functional setting are integral-geometric formulae of Cauchy–Kubota type, valuation property and isoperimetric/Urysohn-like inequalities.

Optimal Battery Dimensioning and Control of a CVT PHEV Powertrain

Abstract: This paper presents convex modeling steps for the problem of optimal battery dimensioning and control of a plug-in hybrid electric vehicle with a continuous variable transmission. The power limits of the internal combustion engine and the electric machine are approximated as convex/concave functions in kinetic energy, whereas their losses are approximated as convex in both kinetic energy and power. An example of minimizing the total cost of ownership of a city bus including a battery wear model is presented. The proposed method is also used to obtain optimal charging power from an infrastructure that is to be designed at the same time the bus is dimensioned.

Log-concavity and strong log-concavity: a review

Abstract: We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

Optimal arrival rate and service rate control of multi-server queues

Abstract: We consider the problem of optimal control of a multi-server queue with controllable arrival and service rates. This study is motivated by its potential application to the design and control of data centers. The cost structure includes customer holding cost which is a non-decreasing convex function of the number of customers in the system, server operating cost which is a non-decreasing convex function of the chosen service rate, and system operating reward which is a non-decreasing concave function of the chosen arrival rate. We formulate the problem as a continuous-time Markov decision process and derive structural properties of the optimal control policies under both discounted cost and average cost criterions.

Rogers-Shephard inequality for log-concave functions

Abstract: In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets.

The numerical solution of Newton's problem of least resistance

Abstract: In this paper we consider Newton's problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in $${\mathbb {R}}^{2}$$ R 2 . We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently. Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in $${\mathbb {R}}^{1}$$ R 1 . Deriving its Euler---Lagrange equation yields a program with two unknowns, which can be solved quickly.

Concave Generalized Flows with Applications to Market Equilibria

Abstract: We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper [Truemper K (1978) Optimal flows in nonlinear gain networks. Networks 8(1):17–36] and by Shigeno [Shigeno M (2006) Maximum network flows with concave gains. Math. Programming 107(3):439–459]. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an e-approximate solution in O(m(mσ+log n)log(MUm/e)) arithmetic operations, where M and U are upper bounds on simple parameters, and σ is the complexity of a value oracle query for the gain functions. This also gives a new algorithm for linear generalized flows, an efficient, purely scaling variant of the Fat-Path algorithm by Goldberg et al. [Goldberg AV, Plotkin SA, Tardos E (1991) Combinatorial algorithms for the generalized circulation problem. Math. Oper. Res. 16(2):351–381], not using any cycle cancellations. We sho...

Convex-concave procedure for weighted sum-rate maximization in a MIMO interference network

Abstract: The weighted sum-rate maximization in a general multiple-input multiple-output (MIMO) interference network has known to be a challenging non-convex problem, mainly due to the interference between different links. In this paper, by exploring the special structure of the sum-rate function being a difference of concave functions, we apply the convex-concave procedure to the weighted sum-rate maximization to handle non-convexity. With the introduction of a certain damping term, we establish the monotonie convergence of the proposed algorithm. Numerical examples show that the introduced damping term slows down the convergence of our algorithm but helps with finding a better solution in the network with high interference. Even though our algorithm has a slower convergence than some existing ones, it has the guaranteed convergence and can handle more general constraints and thus provides a general solver that can find broader applications.

Some new general integral inequalities for h-convex and h-concave functions

Abstract: Abstract. In this paper, we derive new integral inequalities for functions with h -convex and h-concave first derivatives. As a consequence, we give new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are h-convex and h -concave and we point out the results for some special classes of functions.

Optimal Reinsurance Under VaR and CTE Risk Measures When Ceded Loss Function is Concave

Abstract: Cai et al. (2008) explored the optimal reinsurance designs among the class of increasing convex reinsurance treaties under VaR and CTE risk measures. However, reinsurance contracts always involve a limit on the ceded loss function in practice, and thus it may not be enough to confine the analysis to the class of convex functions only. The object of this article is to present an optimal reinsurance policy under VaR and CTE optimization criteria when the ceded loss function is in a class of increasing concave functions and the reinsurance premium is determined by the expected value principle. The outcomes reveal that the optimal form and amount of reinsurance depend on the confidence level p for the risk measure and the safety loading θ for the reinsurance premium. It is shown that under the VaR optimization criterion, the quota-share reinsurance with a policy limit is optima, while the full reinsurance with a policy limit is optima under CTE optimization criterion. Some illustrative examples are provided.

DC Programming Approaches for BMI and QMI Feasibility Problems

Abstract: We propose some new DC (difference of convex functions) programming approaches for solving the Bilinear Matrix Inequality (BMI) Feasibility Problems and the Quadratic Matrix Inequality (QMI) Feasibility Problems. They are both important NP-hard problems in the field of robust control and system theory. The inherent difficulty lies in the nonconvex set of feasible solutions. In this paper, we will firstly reformulate these problems as a DC program (minimization of a concave function over a convex set). Then efficient approaches based on the DC Algorithm (DCA) are proposed for the numerical solution. A semidefinite program (SDP) is required to be solved during each iteration of our algorithm. Moreover, a hybrid method combining DCA with an adaptive Branch and Bound is established for guaranteeing the feasibility of the BMI and QMI. A concept of partial solution of SDP via DCA is proposed to improve the convergence of our algorithm when handling more large-scale cases. Numerical simulations of the proposed approaches and comparison with PENBMI are also reported.

Isoperimetric and Stable Sets for Log-Concave Perturbations of Gaussian Measures

Abstract: Let be an open half-space or slab in R n+1 endowed with a perturbation of the Gaussian measure of the form f(p) := exp(!(p) cjpj 2 ), where c > 0 and ! is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to@ . In this work we follow a variational approach to show that half-spaces perpendicular to @ uniquely minimize the weighted perimeter in among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to@ as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for = R n+1 , which produces in particular the classication of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term! is concave and possibly non-smooth.

Unique reconstruction of convex bodies by projection data via refinements of the Brunn-Minkowski inequality

Abstract: Algorithmic problems in geometry often become tractable with the assumption of convexity. Optimization, volume computation, geometric learning and finding the centroid are all examples of problems that are significantly easier for convex sets. One of the most powerful results in Convex Geometry is the so-called Brunn-Minkowski theorem. It plays a very important role as much in the theoretical framework of geometric inequalities as well as in applied contexts. For instance in crystallography, where it is used to show the mathematical interpretation of the Gibbs-Curie principle (the equilibrium shape of the crystal minimizes the surface energy among all sets of the same volume); or in order to ensure the concavity of the objective function in some optimization problems relative to algorithms for reconstructing a polytope P from surface area measures data. Brunn-Minkowski inequality establishes that the n-th root of the volume of two n-dimensional convex bodies K,E is a concave function, and assuming a common hyperplane projection of K and E (or a common maximal volume section through parallel hyperplanes to a given one), it was proved that the volume itself is concave, i.e., vol(tK+(1-t)E)>tvol(K)+(1-t)vol(E). In this talk we will show, on the one hand, that under the sole assumption that K and E have an equal volume projection (or a common maximal volume section), if the above linear inequality holds with equality for just one value of t in (0,1), then K may be specifically recovered via K=L+E, with L being a segment. We will also discuss that this extra assumption is needed in order to obtain such a characterization. On the other hand, we will show that the expected result of concavity for the k-th root of the volume, when a common projection onto an (n-k)-plane (or a common maximal volume (n-k)-section) is assumed, is not true, by explicitly giving (a family of) convex bodies providing a counterexample for this statement. Nevertheless, other Brunn-Minkowski type inequalities can be derived under this hypothesis. Furthermore, the proofs of these results are constructive in the sense that such bodies can be completely reconstructed by means of the appropriate geometric data in each case.

A simple randomised algorithm for convex optimisation

Abstract: We consider maximising a concave function over a convex set by a simple randomised algorithm The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron

The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height

Abstract: For the minimal graph with strictly convex level sets, we find an auxiliary function to study the Gaussian curvature of the level sets. We prove that this curvature function is a concave function with respect to the height of the minimal surface while this auxiliary function is almost sharp when the minimal surface is the catenoid.

A generalisation of Mirsky's singular value inequalities

Abstract: We prove an f-version of Mirsky's singular value inequalities for differences of matrices. This f-version consists in applying a positive concave function f, with f(0)=0, to every singular value in the original Mirsky inequalities.

A sharp Blaschke–Santaló inequality for \alpha -concave functions

Abstract: We define a new transform on $$\alpha $$ -concave functions, which we call the $$\sharp $$ -transform. Using this new transform, we prove a sharp Blaschke–Santalo inequality for $$\alpha $$ -concave functions, and characterize the equality case. This extends the known functional Blaschke–Santalo inequality of Artstein-Avidan, Klartag and Milman, and strengthens a result of Bobkov. Finally, we prove that the $$\sharp $$ -transform is a duality transform when restricted to its image. However, this transform is neither surjective nor injective on the entire class of $$\alpha $$ -concave functions.

Hessian matrices of penalty functions for solving constrained-optimization problems

Abstract: This paper deals with the Hessian matrices of penalty functions, evaluated at their minimizing point. It is concerned with the condition number of these matrices for small values of a controlling parameter. At the end of the paper a comparison is made between different types of penalty functions on the grounds of the results obtained. 1. Classification of penalty-function techniques Throughout this paper we shall be concerned with the problem minimize f(x) subject t~ } g,(x) ~ 0; 1 = I, ... ,m, where x denotes an element of the n-dimensional vector space En. We shall be assuming that the functions J, -gl> ... , -gm are convex and twice differentiable with continuous second-order partial derivatives on an open, convex subset V of En. The constraint set (1.1) R={xlg,(x);;:;::O; i=I, ... ,m} (1.2) is a bounded subset of V. The interior Ro of R is non-empty. We consider a number of penalty-function techniques for solving problem (1.1). One can distinguish two classes, both of which have been referred to by expressive names. The interior-point methods operate in the interior Ro of R. The penalty function is given by m Br(x) = f(x) r ~ cp[g,(x)], '=1 (1.3) where cp is a concave function of one variable, say y. Its derivative tp' reads cp'(y) = y-V (l.4) with a positive, integer '11. A point x(r) minimizing (1.3) over Ro 'exists then for "any r > 0, Any convergent sequence {x(rk)}' where {rk} is a monotonie, decreasing null sequence as k --* 00, converges to a solution of (1.1). The exterior-point methods or outside-in methods present an approach to a minimum solution from outside the constraint set. The' general form of the HESSIAN MATRICES OF PENALTY FUNCTIONS penalty function is given by m Lr(x) = f(x) r-1 ~ lp[g,(x)], '=1 where 11' is a concave function of one variable y, such that {° for y < 0, 1p(y) = co(y) for y::::;; 0. The derivative co' of co is given by co'(y) = (-y)V. Let z(r) denote a point minimizing (1.5) over En. Any convergent sequence {zerk)}, where {rk} denotes again a monotonie, decreasing null sequence, converges to a solution of (1.1). It will be convenient to extend the terminology that we have been using in previous papers. Following Murray S) we shall refer to interior-point penalty functions of the type (1.3) as barrier functions. The exterior-point penalty functions (1.5) will briefly be indicated as loss functions, a name which has also been used by Fiacco and McCormick 1). Furthermore, we introduce a classification based on the behaviour of the functions tp' and co' in a neighbourhood of y = O. A barrier function is said to be of order 'JI if the function tp' has a pole of order 'JI at y = o. Similarly, a loss function is of order 'JI if the function co' has a zero of order 'JI at y = o. 2. Conditioning An intriguing point is the choice of a penalty function for numerical purposes. We shall not repeat here all the arguments supporting the choice of the firstorder penalty functions for computational purposes. Our concern is an argument which has been introduced only by Murray S), namely the question of "conditioning". This is a qualification referring to the Hessian matrix of a penalty function. The motivation for such a study is the idea that failures of (second-order) unconstrained-minimization techniques may be due to illconditioning of the Hessian matrix at some iteration points. Throughout this paper it is tacitly assumed that penalty functions are strictly convex so that they have a unique minimizing point in their definition area. We shall primarily be concerned with the Hessian matrix of penalty functions at the minimizing point. In what follows we shall refer to it as the principal Hessian matrix. The reason will be clear. In a neighbourhood ofthe minimizing point a useful approximation of a penalty function is given by a quadratic function, with the principal Hessian matrix as the coefficient matrix of the quadratic term. It is therefore reasonable to assume that unconstrained mini323

The interpolation method for random graphs with prescribed degrees

Abstract: We consider large random graphs with prescribed degrees, such as those generated by the configuration model. In the regime where the empirical degree distribution approaches a limit $\mu$ with finite mean, we establish the systematic convergence of a broad class of graph parameters that includes in particular the independence number, the maximum cut size and the log-partition function of the antiferromagnetic Ising and Potts models. The corresponding limits are shown to be Lipschitz and concave functions of $\mu$. Our work extends the applicability of the celebrated interpolation method, introduced in the context of spin glasses, and recently related to the fascinating problem of right-convergence of sparse graphs.

Weighted Hardy-type inequalities on the cone of quasi-concave functions

Abstract: The paper is devoted to the study of weighted Hardy-type inequalities on the cone of quasi-concave functions, which is equivalent to the study of the boundedness of the Hardy operator between the L ...

A Utility Equivalence Theorem for Concave Functions

Abstract: Given any two sets of independent non-negative random variables and a non-decreasing concave utility function, we identify sufficient conditions under which the expected utility of sum of these two sets of variables is (almost) equal. We use this result to design a polynomial-time approximation scheme (PTAS) for utility maximization in a variety of risk-averse settings where the risk is modeled by a concave utility function. In particular, we obtain a PTAS for the asset allocation problem for a risk-averse investor as well as the risk-averse portfolio allocation problem.

Exploiting the convex-concave penalty for tracking: A novel dynamic reweighted sparse Bayesian learning algorithm

Abstract: We propose a novel dynamic reweighted l 2 (DRl 2 ) algorithm in the regime of dynamic compressive sensing. Our analysis shows that aiming to solve a Type II optimization problem, DRl 2 is effectively minimizing a `convex-concave' penalty in the coefficients that transitions from a convex region to a concave function using knowledge of past estimations. DRl 2 thus provides superior reconstruction performance compared with state-of-the-art dynamic CS algorithms.

Strategy-Proof Package Assignment

Abstract: We examine the strategy-proof allocation of multiple divisible and indivisible resources; an application is the assignment of packages of tasks, workloads, and compensations among the members of an organization. We find that any allocation mechanism obtained by maximizing a separably concave function over a polyhedral extension of the set of Pareto-efficient allocations is strategy-proof. Moreover, these are the only strategy-proof and unanimous mechanisms satisfying a coherence property and responding well to changes in the availability of resources. These mechanisms generalize the parametric rationing mechanisms (Young, 1987), some of which date back to the Babylonian Talmud.

Theory of locally concave functions and its applications to sharp estimates of integral functionals

Abstract: We prove a duality theorem the computation of certain Bellman functions is usually based on. As a byproduct, we obtain sharp results about the norms of monotonic rearrangements. The main novelty of our approach is a special class of martingales and an extremal problem on this class, which is dual to the minimization problem for locally concave functions.