Nondifferentiable Points Research Articles

Local Pareto approximation for multi-objective optimization

The design process of complex systems often resorts to solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically, this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the objective function space. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution for sensitivity analysis. In this article, new linear and quadratic local approximations of the Pareto surface are derived and compared to existing formulas. The case of non-differentiable Pareto points (solutions) in the objective space is also analysed. The concept of a local quick Pareto analyser based on local sensitivity analysis is proposed. This Pareto analysis provides a quantitative insight into the relation between variations of the different objective functions under constraints. A few examples are considered to illustrate the concept and its advantages.

Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization

Conditional Value-at-Risk (CVaR) is a portfolio evaluation function having appealing features such as sub-additivity and convexity. Although the CVaR function is nondifferentiable, scenario-based CVaR minimization problems can be reformulated as linear programs (LPs) that afford solutions via widely-used commercial softwares. However, finding solutions through LP formulations for problems having many financial instruments and a large number of price scenarios can be time-consuming as the dimension of the problem greatly increases. In this paper, we propose a two-phase approach that is suitable for solving CVaR minimization problems having a large number of price scenarios. In the first phase, conventional differentiable optimization techniques are used while circumventing nondifferentiable points, and in the second phase, we employ a theoretically convergent, variable target value nondifferentiable optimization technique. The resultant two-phase procedure guarantees infinite convergence to optimality. As an optional third phase, we additionally perform a switchover to a simplex solver starting with a crash basis obtained from the second phase when finite convergence to an exact optimum is desired. This three phase procedure substantially reduces the effort required in comparison with the direct use of a commercial stand-alone simplex solver (CPLEX 9.0). Moreover, the two-phase method provides highly-accurate near-optimal solutions with a significantly improved performance over the interior point barrier implementation of CPLEX 9.0 as well, especially when the number of scenarios is large. We also provide some benchmarking results on using an alternative popular proximal bundle nondifferentiable optimization technique.

Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles

Given a bounded domain Ω \Omega in R 2 \mathbb {R}^2 with smooth boundary, the cut locus Σ ¯ \overline \Sigma is the closure of the set of nondifferentiability points of the distance d d from the boundary of Ω \Omega . The normal distance to the cut locus, τ ( x ) \tau (x) , is the map which measures the length of the line segment joining x x to the cut locus along the normal direction D d ( x ) Dd(x) , whenever x ∉ Σ ¯ x\notin \overline \Sigma . Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of Ω \Omega is of class C 2 , 1 C^{2,1} . Our main result is the global Hölder regularity of τ \tau in the case of a domain Ω \Omega with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called regular conjugate points is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain Ω \Omega . The above regularity result for τ \tau is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.

On sets of non-differentiability of Lipschitz and convex functions

We observe that each set from the system $\widetilde{\mathcal A}$ (or even $\widetilde{\mathcal{C}}$) is $\Gamma $-null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on ${\mathbb{R}}^n$ is $\sigma $-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented.

Locus of boundary crisis: Expect infinitely many gaps

Boundary crisis is a mechanism for destroying a chaotic attractor when one parameter is varied. In a two-parameter setting the locus of the boundary crisis is associated with curves of homoclinic or heteroclinic bifurcations of periodic saddle points. It is known that this locus has nondifferentiable points. We show here that the locus of boundary crisis is far more complicated than previously reported. It actually contains infinitely many gaps, corresponding to regions (of positive measure) where attractors exist.

Phase transitions for the multifractal analysis of self-similar measures

We are interested in the multifractal analysis of a class of self-similar measures with overlaps. This class, for which we obtain explicit formulae for the Lq-spectrum, τ(q), as well as the singularity spectrum f(α), is sufficiently large to point out new phenomena in the multifractal structure of self-similar measures. We show that, unlike in the classical quasi-Bernoulli case, the Lq-spectrum, τ(q), of the measures studied can have an arbitrarily large number of non-differentiability points (phase transitions). These singularities occur only for the negative values of q and yield to measures that do not satisfy the usual multifractal formalism. The weak quasi-Bernoulli property is the key point of most of the arguments.

A New Approach for Multiple-Response Optimization

The desirability of a response that increases in a linear manner from the lower specification limit to the target value and decreases in a linear manner from the target value to the upper specification limit results in a breakpoint (nondifferentiable point) at the target value of its desirability function. Thus, modified desirability functions were proposed to solve this breakpoint problem so that the generalized reduced gradient can be applied. The modified desirability functions require the fitting of a quartic polynomial in the vicinity of the target. However, the fitting of the quartic polynomial needs to satisfy five conditions and involves numerous computations. Therefore, an approach with less computations and yet that is simple to implement is proposed in this article. This approach is applicable when the lower specification limit, the target value, and the upper specification limit of a response are available. In addition, this approach also allows practitioners to assign different degrees of importance to the responses. An example from the original paper of modified desirability functions is used to illustrate this approach.

On singularities of value function for Bolza optimal control problem

We study the set of points of nondifferentiability, called the singular set, of the value function of a Bolza optimal control problem. The value function is a viscosity solution to an associated Hamilton–Jacobi equation. The method of characteristics associates to this equation a Hamiltonian system, that in turn can be used to study the propagation of singularities of the value function. In particular, we obtain an extension of the Rankine–Hugoniot type condition, which is well-known in the conservation law theory.

Lipschitz functions with unexpectedly large sets of nondifferentiability points

It is known that every Gδ subset E of the plane containing a dense set of lines, even if it has measure zero, has the property that every real‐valued Lipschitz function on ℝ2 has a point of differentiability in E. Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct a Gδ set E ⊂ ℝ2 containing a dense set of lines for which there is a pair of real‐valued Lipschitz functions on ℝ2 having no common point of differentiability in E, and there is a real‐valued Lipschitz function on ℝ2 whose set of points of differentiability in E is uniformly purely unrectifiable.

Identification of Linear Error-Models with Projected Dynamical Systems

Linear error models are an integral part of several parameter identification methods for feedforward and feedback control systems and lead in connection with the L 2-norm to a convex distance measure which has to be minimised for identification purposes. The parameters are hereby often subject to specific restrictions whose intersections span a convex solution set with non-differentiability points on its boundary. For solving these well conditioned problems on-line the paper formulates the solution of the bounded convex minimisation problem as a stable equilibrium set of a proper system of differential equations. The vector field of the corresponding system of differential equations is based on a projection of the negative gradient of the distance measure. A general drawback of this approach is the discontinuous right-hand side of the differential equation caused by the projection transformation. The consequence are difficulties for the verification of the existence, uniqueness and stability of a solution trajectory. Therefore the first subject of this paper is the derivation of an alternative formulation of the projected dynamical system, which exhibits, in contrast to the original formulation, a continuous right-hand side and is thus accessible to conventional analysis methods. For this purpose the multi-dimensional stop operator is used and the existence, uniqueness and stability properties of the solution trajectories are established. The second part of this paper deals with the numerical integration of the projected dynamical system which is used for an implementation of the identification method on a digital signal processor for example. To demonstrate the performance the application of this on-line identification method to the hysteretic filter synthesis with the modified Prandtl-Ishlinskii approach is presented in the last part of this paper.

Understanding atomic processes in terms of Coulomb singularities

Many aspects of high-energy atomic processes can be described in terms of singularities of a many-body Hamiltonian using the generalized asymptotic Fourier transform (AFT) theory. The study of matrix elements in different kinematic regimes is related to the study of singularities (points of nondifferentiability) of the wave functions and the e-γ interaction. These singularities reflect the singularities of the many-body Hamiltonian. We illustrate the principles of the AFT approach in the simple example of photoabsorption by the electron bound in a potential with a Coulomb singularity. We exhibit two general results that are important for any many-body system: (1) the quality of approximate results in different forms (“gages”) depends on the quality of the description of the wave functions in the vicinity of singularities, and (2) due to the character of the Coulomb singularity, photoabsorption cross sections converge slowly to their asymptotic form as the energy increases. However, the slowly converging behavior of these cross sections is due to one common factor (the Stobbe factor), which can be obtained analytically in terms of the characterization of the vicinity of the singularity. The common Stobbe factor explains why ratios of cross sections converge more rapidly than the cross sections themselves.

Parallel Evolutionary Optimization of Multibody Systems with Application to Railway Dynamics

The optimization of multibody systems usually requires many costlycriteria computations since the equations of motion must be evaluated bynumerical time integration for each considered design. For activelycontrolled or flexible multibody systems additional difficulties ariseas the criteria may contain non-differentiable points or many localminima. Therefore, in this paper a stochastic evolution strategy is used incombination with parallel computing in order to reduce the computationtimes whilst keeping the inherent robustness. For the parallelization amaster-slave approach is used in a heterogeneous workstation/PCcluster. The pool-of-tasks concept is applied in order to deal withthe frequently changing workloads of different machines in the cluster. In order to analyze the performance of the parallel optimizationmethod, the suspension of an ICE passenger coach, modelled as an elasticmultibody system, is optimized simultaneously with regard to severalcriteria including vibration damping and a criterion related to safetyagainst derailment. The iterative and interactive nature of a typicaloptimization process for technical systems is emphasized.

Gâteaux Differentiability of Lipschitz Functions via Directional Derivatives

Let \(X\) be a separable Banach space, \(Y\) a Banach space and \(f:X \to Y\) a Lipschitz function. We show that the set of all G\^ ateaux non-differentiability points at which \(f\) has all one-sided or two-sided directional derivatives can be covered by (special subsets of) Lipschitz surfaces of codimension 1 or codimension 2, respectively. Further results indicate that these results are close to the best possible ones. Our results are new also for Lipschitz functions \(\R^n \to \R\); for these functions G\^ ateaux differentiability is the classical (total) differentiability.

Non-differentiability of devil's staircases and dimensions of subsets of Moran sets

Let C be the homogeneous Cantor set invariant for x→ax and x→1−a+ax. It has been shown by Darst that the Hausdorff dimension of the set of non-differentiability points of the distribution function of uniform measure on C equals (dimHC)2 = (log 2/log a)2. In this paper we generalize the essential ingredient of the proof of this result. Let Ω = {0, 1, …, r}. Let F be a Moran set associated with {0 < ai < 1, i ∈ Ω} and Ωw = Ω×Ω×⃛. Let ø be the associated coding map from Ωe onto F. Fix a non-empty set Γ ⊆ Ω with Γ≠Ø and let z(σ, n) denote the position of the nth occurrence of the elements of Γ in σ ∈ Ωw.

Closed loops of liquid–liquid immiscibility predicted by semi-empirical cubic equations of state and classical mixing rules

The present study demonstrates that semi-empirical cubic EOSs, which include temperature functionalities such as that of Soave and van der Waals classical mixing rules, predict the behavior around the mathematical double points of the second kind. For the first time these results allow one to fit Type VI behavior in real water solutions, that include both first and second hypercritical points, or the absence of each of these points. Although the predicted data match the experimental data, their genesis is explained by fundamental numerical pitfalls, namely non-physical and non-differentiable breaking points that are characteristic for several empirical temperature functionalities. Hence, such results cannot represent the physical nature of the real systems. Although these pitfalls may take place outside the range of practical significance for pure compounds, they may strongly affect the predictions of the mixtures even at ordinary conditions. Therefore it is shown once again that all parts of the thermodynamic phase space predicted by the EOSs are closely inter-related.

The Numerical Range of Self-Adjoint Quadratic Matrix Polynomials

Matrix polynomials of the form $P(\lambda)=I\lambda^2+A_1\lambda+A_0$ (where A0 and A1 are n × n Hermitian matrices and $\lambda$ is a complex variable) arise in many applications. The numerical range of such a polynomial is \[ W(P) = \{\lambda \in \mathbb{C} : x^*P(\lambda)x = 0 \mbox{ for some nonzero } x \in \mathbb{C}^n \} \] and it always contains the spectrum of $P(\lambda)$, i.e., the set of zeros of $\textup{det}P(\lambda)$. Properties of the numerical range are developed in detail, taking advantage of the close connection between W(P) and the classical numerical range (field of values) of the (general) complex matrix A:=A0+iA1. Eigenvalues and nondifferentiable points on the boundary are examined and a procedure for the numerical determination of W(P) is presented and used for several illustrations. Some extensions of the theory to more general polynomials $P(\lambda)$ are also discussed, as well as special cases describing vibrating systems.

A nonsmooth Newton method for elastoplastic problems

In this work we reformulate the incremental, small strain, J 2-plasticity problem with linear kinematic and nonlinear isotropic hardening as a set of unconstrained, nonsmooth equations. The reformulation is done using the minimum function. The system of equations obtained is piecewise smooth which enables Pang's Newton method for B-differentiable equations to be used. The method proposed in this work is compared with the familiar radial return method. It is shown, for linear kinematic and isotropic hardening, that this method represents a piecewise smooth mapping as well. Thus, nonsmooth Newton methods with proven global convergence properties are applicable. In addition, local quadratic convergence (even to nondifferentiable points) of the standard implementation of the radial return method is established. Numerical tests indicate that our method is as efficient as the radial return method, albeit more sensitive to parameter changes.

Efficient optimization by modifying the objective function: applications to timing-driven VLSI layout

When minimizing a given objective function is challenging because of, for example, combinatorial complexity or points of nondifferentiability, one can apply more efficient and easier-to-implement algorithms to modified versions of the function. In the ideal case, one can employ known algorithms for the modified function that have a thorough theoretical and empirical record and for which public implementations are available. The main requirement here is that minimizers of the objective function do not change much through the modification, i.e., the modification must have a bounded effect on the quality of the solution. Review of classic and recent placement algorithms suggests a dichotomy between approaches that either: (a) heuristically minimize potentially irrelevant objective function (e.g., VLSI placement with quadratic wirelength) motivated by the simplicity and speed of a standard minimization algorithm; or (b) devise elaborate problem-specific minimization heuristics for more relevant objective functions (e.g., VLSI placement with linear wirelength). Smoothness and convexity of the objective functions typically enable efficient minimization. If either characteristic is not present in the objective function, one can modify and/or restrict the objective to special values of parameters to provide the missing properties. After the minimizers of the modified function are found, they can be further improved with respect to the original function by fast local search using only function evaluations. Thus, it is the modification step that deserves most attention. In this paper, we approximate convex nonsmooth continuous functions by convex differentiable functions which are parameterized by a scalar /spl beta/>0 and have convenient limit behavior as /spl beta//spl rarr/0. This allows the use of Newton-type algorithms for minimization and, for standard numerical methods, translates into a tradeoff between solution quality and speed. We prove that our methods apply to arbitrary multivariate convex piecewise-linear functions that are widely used in synthesis and analysis of electrical networks. The utility of our approximations is particularly demonstrated for wirelength and nonlinear delay estimations used by analytical placers for VLSI layout, where they lead to more solvable problems than those resulting from earlier comparable approaches [29]. For a particular delay estimate, we show that, while convexity is not straightforward to prove, it holds for a certain range of parameters, which, luckily, are representative of real-world technologies.

Frequency Locking on the Boundary of the Barycentre Set

We consider the doubling map T : z Z2 of the circle. For each T-invariant probability measure μ we define its barycentre b(μ) = ∫S1 Z dμ(z), which describes its average weight around the circle. We study the set Ω of all such barycentres, a compact convex set with nonempty interior. Its boundary ∂Ω has a countable dense set of points of nondifferentiability, the worst possible regularity for the boundary of a convex set. We explain this behaviour in terms of the frequency locking of rotation numbers for a certain class of invariant measures, each supported on the closure of a Sturmian orbit.

CONTROL OPTIMIZATION F MULTIBODY SYSTEMS SING POINT- AND PIECEWISE EFINED OPTIMIZATION CRITERIA

A flexible and simplified method for the formulation of criteria for the optimization of muliibody systems is presented using point- and piecewise defined functions. These functions may lead to non-differentiable points of the optimization criteria and discontinuous sensitivities. Their computation and influence on the convergence behavior of an SQP-algorithm is examined. The sensitivities are computed using the Automatic Differentiation method, which can easily be adapted to a wide variety of computer-evaluated gradients. The application of point- and piecewise defined optimization criteria as well as the discontinuous sensitivities are shown for the example of an actively damped truss on a space station.