Showing papers on "Concave function published in 2001"

An alternative interpolation scheme for minimum compliance topology optimization

Abstract: We consider the discretized zero-one continuum topology optimization problem of finding the optimal distribution of two linearly elastic materials such that compliance is minimized. The geometric complexity of the design is limited using a constraint on the perimeter of the design. A common approach to solve these problems is to relax the zero-one constraints and model the material properties by a power law which gives noninteger solutions very little stiffness in comparison to the amount of material used. We propose a material interpolation model based on a certain rational function, parameterized by a positive scalar q such that the compliance is a convex function when q is zero and a concave function for a finite and a priori known value on q. This increases the probability to obtain a zero-one solution of the relaxed problem.

Filterbank optimization with convex objectives and the optimality of principal component forms

Abstract: This paper proposes a general framework for the optimization of orthonormal filterbanks (FBs) for given input statistics. This includes as special cases, many previous results on FB optimization for compression. It also solves problems that have not been considered thus far. FB optimization for coding gain maximization (for compression applications) has been well studied before. The optimum FB has been known to satisfy the principal component property, i.e., it minimizes the mean-square error caused by reconstruction after dropping the P weakest (lowest variance) subbands for any P. We point out a much stronger connection between this property and the optimality of the FB. The main result is that a principal component FB (PCFB) is optimum whenever the minimization objective is a concave function of the subband variances produced by the FB. This result has its grounding in majorization and convex function theory and, in particular, explains the optimality of PCFBs for compression. We use the result to show various other optimality properties of PCFBs, especially for noise-suppression applications. Suppose the FB input is a signal corrupted by additive white noise, the desired output is the pure signal, and the subbands of the FB are processed to minimize the output noise. If each subband processor is a zeroth-order Wiener filter for its input, we can show that the expected mean square value of the output noise is a concave function of the subband signal variances. Hence, a PCFB is optimum in the sense of minimizing this mean square error. The above-mentioned concavity of the error and, hence, PCFB optimality, continues to hold even with certain other subband processors such as subband hard thresholds and constant multipliers, although these are not of serious practical interest. We prove that certain extensions of this PCFB optimality result to cases where the input noise is colored, and the FB optimization is over a larger class that includes biorthogonal FBs. We also show that PCFBs do not exist for the classes of DFT and cosine-modulated FBs.

Differentiability of convex envelopes

Abstract: We prove that the convex envelope of a differentiable, or C 1, α -function f is C 1 , or C 1, α respectively, provided only that the function satisfies the very mild growth condition that f ( x ) tends to +∞ if | x | does so.

Exact number of solutions of a class of two-point boundary value problems involving concave and convex nonlinearities☆

Abstract: In this paper, exact number of solutions are obtained for a class of two-point boundary value problems involving concave and convex nonlinearities. Moreover, some properties of the solutions are studied in details.

On the interpolation constant for Orlicz spaces

Abstract: The authors discuss the interpolation from Lebesgue spaces into Orlicz spaces, with special attention to the interpolation constant $C$. The authors give some estimates for $C$.

Monotonicity and convexity properties of matrix Riccati equations

Abstract: Using a Frechet-derivative-based approach some monotonicity, convexity/concavity and comparison results concerning strictly unmixed solutions of continuous- and discrete-time algebraic Riccati equations are obtained; it turns out that these solutions are isolated and smooth functions of the input data. Similarly, it is proved that the solutions of initial value problems for both Riccati differential and difference equations are smooth and monotonic functions of the input data and of the initial value. They are also convex or concave functions with respect to certain matrix coefficients.

Some Observations on Equation-Based Rate Control

Abstract: We consider one aspect of the general problem of unicast equation based rate control in the Internet, which we formulate as follows. When a so called ``loss-event" occurs, a data source updates its sending rate by setting it to $f(\hat{p_n})$, where $\hat{p}_n$ is an estimate of $\overline{p}$, the rate of loss-events. Function $f$ (the target loss-throughput function) defines the objective of the control method: we would like that the throughput $\overline{x}$, attained by the source, satisfies the equation $\overline{x}\leq f(\overline{p})$. If so, we say that the control is conservative. In the Internet, function $f$ is obtained by analyzing the dependency of throughput versus the rate of loss-events for a real TCP source. A non-TCP source which implements a control system as we describe is said to be TCP-friendly if the control is conservative. In this paper, we examine whether such a control system is conservative. We first consider a simple stochastic model which assumes that the intensity of the loss-events is proportional to the current sending rate. We show that, for this model, the control is always conservative if $f(p)$ is a concave function of $1/p$; otherwise this may not be true. Then we consider a second model where the loss-event inter-arrival times is an exogeneous stationary random process. We show that, for this second model, there exist statistics of the loss-event inter-arrival times such that the control is non-conservative, even if $f(p)$ is a concave function of $1/p$. We validate our analytical results with simulations. Another aspect of unicast equation-based rate control in the Internet is the influence of the variability of round-trip times, which is not analyzed in this paper. KEYWORDS: Equation-based, Rate control, TCP-friendliness, Internet, Stochastic recurrence, Autoregressive process, Markov modulated process, Non-linear system, Estimation, Palm expectation

Optimization over equilibrium sets

Abstract: We introduce a certain notion of equilibrium points, which constitute a generalization of Pareto efficient points, and we propose a branch-and-bound method to maximize a concave function over the set of all equilibrium points of a given set

On the Convergence of Cone Splitting Algorithms with ω-Subdivisions

Abstract: We present a convergence proof of the Tuy cone splitting algorithm with a pure ?-subdivision strategy for the minimization of a concave function over a polytope. The key idea of the convergence proof is to associate with the current hyperplane a new hyperplane that supports the whole polytope instead of only the portion of it contained in the current cone. A branch-and-bound variant of the algorithm is also discussed.

On Tuy’s 1964 Cone Splitting Algorithm for Concave Minimization

Abstract: Since the work of Zwart, it is known that cycling may occur in the cone splitting algorithm proposed by Thy in 1964 to minimize a concave function over a polytope. In this paper, we show that despite this fact, Tuy’s algorithm is convergent in the sense that it always finds an optimal solution. This result also holds for a variant of Tuy’s algorithm proposed by Gallo, in which a cone is split into a smaller subset of subcones (in term of inclusion). As shown by an example, this variant may also cycle. The transformation of these two algorithms into finite step procedures is discussed.

Multiple sequence alignment using the quasi-concave function optimization based on the DIALIGN combinatorial structures

Abstract: Multiple sequence alignment is usually considered as an optimization problem, which has a statistical and a structural component. It is known that in the problem of protein sequence alignment a processed sample is too small and not representative in the statistical sense though this information can be sufficient if an appropriate structural model is used. In order to utilize this information a new structural description of the pairwise alignment results union has been developed. It is shown that if the structure is restored then Multiple Sequence Alignment is achieved. Introduced structure represents the set of local maximums of quasi-concave set function on a lower semi lattice, which in turn is a union of the set-theoretical intervals. This union is a set of the consistent subsets of diagonals, introduced by B. Morgenstern, A. Dress, and T. Werner (1996). Algorithm for local maximums search on proposed structure has been developed. It consists of an alternation of the Forward and Backward passes. The Backward pass in this algorithm is a rigorous while the Forward pass is based on heuristics. Multiple alignment of 5 protein sequences are used as an illustration of the proposed algorithm.

Supply management with concave cost

Abstract: This paper is divided in two parts. In the first part, we consider the case where several providers feed a single manufacturing unit periodically for certain specified demand. The cost incurred when a provider feeds the manufacturing unit is a concave function of the quantity delivered. Furthermore, each provider either delivers a quantity that lies between a minimum and maximum value or do not deliver any thing. We are introducing some properties of optimal solution and derive a heuristic algorithm from these properties. A numerical example illustrates this approach. In the second part of the paper, we consider the case where several providers feed periodically several manufacturing units. A heuristic algorithm is proposed. The results provided by this algorithm are compared with the optimal solution for the costs that are linear for a strictly positive quantity and equal to zero otherwise.