René Henrion

N-way principal component analysis theory, algorithms and applications

Abstract Due to sophisticated experimental designs and to modern instrumental constellations the investigation of N -dimensional (or N -way or N -mode) data arrays is attracting more and more attention. Three-dimensional arrays may be generated by collecting data tables with a fixed set of objects and variables under different experimental conditions, at different sampling times, etc. Stacking all the tables along varying conditions provides a cubic arrangement of data. Accordingly the three index sets or modes spanning a three-way array are called objects, variables and conditions. In many situations of practical relevance even higher-dimensional arrays have to be considered. Among numerous extensions of multivariate methods to the three-way case the generalization of principal component analysis (PCA) has central importance. There are several simplified approaches of three-way PCA by reduction to conventional PCA. One of them is unfolding of the data array by combining two modes to a single one. Such a procedure seems reasonable in some specific situations like multivariate image analysis, but in general combined modes do not meet the aim of data reduction. A more advanced way of unfolding which yields separate component matrices for each mode is the Tucker 1 method. Some theoretically based models of reduction to two-way PCA impose some specific structure on the array. A proper model of three-way PCA was first formulated by Tucker (so-called Tucker 3 model among other proposals). Unfortunately the Tucker 1 method is not optimal in the least squares sense of this model. Kroonenberg and De Leeuw demonstrated that the optimal solution of Tuckers model obeys an interdependent system of eigenvector problems and they proposed an iterative scheme (alternating least squares algorithm) for solving it. With appropriate notation Tuckers model as well as the solution algorithm are easily generalized to the N -way case (N > 3). There are some specific aspects of three-way PCA, such as complicated ways of data scaling or interpretation and simple-structure-transformation of a so-called core matrix, which make it more difficult to understand than classical PCA. An example from water chemistry serves as an illustration. Additionally, there is an application section demonstrating several rules of interpretation of loading plots with examples taken from environmental chemistry, analysis of complex round robin tests and contamination analysis in tungsten wire production.

Calmness of constraint systems with applications

The paper is devoted to the analysis of the calmness property for constraint set mappings. After some general characterizations, specific results are obtained for various types of constraints, e.g., one single nonsmooth inequality, differentiable constraints modeled by polyhedral sets, finitely and infinitely many differentiable inequalities. The obtained conditions enable the detection of calmness in a number of situations, where the standard criteria (via polyhedrality or the Aubin property) do not work. Their application in the framework of generalized differential calculus is explained and illustrated by examples associated with optimization and stability issues in connection with nonlinear complementarity problems or continuity of the value-at-risk.

On the Calmness of a Class of Multifunctions

The paper deals with the calmness of a class of multifunctions in finite dimensions. Its first part is devoted to various conditions for calmness, which are derived in terms of coderivatives and subdifferentials. The second part demonstrates the importance of calmness in several areas of nonsmooth analysis. In particular, we focus on nonsmooth calculus and solution stability in mathematical programming and in equilibrium problems. The derived conditions find a number of applications there.

Convexity of chance constraints with independent random variables

Abstract We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.

Second-Order Analysis of Polyhedral Systems in Finite and Infinite Dimensions with Applications to Robust Stability of Variational Inequalities

This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimensions.

Metric regularity and quantitative stability in stochastic programs with probabilistic constraints

Introducing probabilistic constraints leads in general to nonconvex, nonsmooth or even disconti- nuous optimization models. In this paper, necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis. The conditions apply to fairly general constraints and extend earlier work in this direction. Further, a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied in order to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations. This is used to derive bounds for the deviation of solution sets when the probability measure is replaced by empirical estimates.

On constraint qualifications

The linear independence constraint qualification (LICQ) and the weaker Mangasarian-Fromovitz constraint qualification (MFCQ) are well-known concepts in nonlinear optimization. A theorem is proved suggesting that the set of feasible points for which MFCQ essentially differs from LICQ is small in a specified sense. As an auxiliary result, it is shown that, under MFCQ, the constraint set (even in semi-infinite optimization) is locally representable in epigraph form.

Subdifferential Conditions for Calmness of Convex Constraints

We study subdifferential conditions of the calmness property for multifunctions representing convex constraint systems in a Banach space. Extending earlier work in finite dimensions [R. Henrion and J. Outrata, J. Math. Anal. Appl., 258 (2001), pp. 110--130], we show that, in contrast to the stronger Aubin property of a multifunction (or metric regularity of its inverse), calmness can be ensured by corresponding weaker constraint qualifications, which are based only on boundaries of subdifferentials and normal cones rather than on the full objects. Most of the results can be immediately interpreted in the context of error bounds.

Optimization of a continuous distillation process under random inflow rate

Abstract The paper deals with a continuous distillation process under stochastic rate of inflows collected in a feed tank. The aim of analysis is to find a robust control of extracting feed from the tank over a certain time horizon such that—without knowledge of future realizations of the inflow rate—some level constraints in the feed tank will be met with high probability. This approach relies on formulating and numerically treating probabilistic constraints. The inflow rate is considered as a stochastic process for which two basically different model assumptions are made: the first model assumes a Gaussian process, and thus reflects the superposition of many independent elementary inflows; the second model treats maybe the simplest case of a single elementary inflow profile, namely rectangular inflows with fixed rate and duration but stochastic starting time. Numerical results illustrating both assumptions are presented, and advantages over the simple anticipation of nominal inflow profiles are highlighted.

A new criterion for simple-structure transformations of core arrays in N-way principal components analysis

Abstract Among the possible (orthogonal) transformations of core arrays in N-way principal components analysis (PCA), the conventional approach of body diagonalization turns out not to provide the simplest structure (in the sense of minimizing the number of significant entries). As an alternative, the maximization of the variance-of-squared core entries is proposed. Both criteria are equivalent in a two-way constellation but may differ markedly for N≥3. Actually, using the variance criterion may provide more insight into the rank structure of the given data, and it is also easily applied to general rectangular core arrays. In order to clarify the relation between body diagonality and variance-of-squares, we prove the following main result of the paper: If some cubic N-way core array can be transformed to exact body diagonality, then the same transformation yields maximum variance-of-squared entries. This result implies the equivalence in the two-way case mentioned above. A solution algorithm is formulated and illustrated with a small numerical example. The application to data examples from environmental chemistry and chromatographic analysis is briefly discussed.