We investigate quasi-Newton methods for minimizing a strongly convex quadratic function which is subject to errors in the evaluation of the gradients. In particular, we focus on computing search directions for quasi-Newton methods that all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. The BFGS quasi-Newton method may be seen as an ideal method in exact arithmetic and is empirically known to behave very well on a quadratic problem subject to small errors. We investigate large-error scenarios, in which the expected behavior is not so clear. We consider memoryless methods that are less expensive than the BFGS method, in that they generate low-rank quasi-Newton matrices that differ from the identity by a symmetric matrix of rank two. In addition, a more advanced model for generating the search directions is proposed, based on solving a chance-constrained optimization problem. Our numerical results indicate that for large errors, such a low-rank memoryless quasi-Newton method may perform better than a BFGS method. In addition, the results indicate a potential edge by including the chance-constrained model in the memoryless quasi-Newton method.

Support vector machines (SVM) is one of the well known supervised machine learning model. The standard SVM models are dealing with the situation where the exact values of the data points are known. This paper studies the SVM model when the data set contains uncertain or mislabelled data points. To ensure the small probability of misclassification for the uncertain data, a chance constrained conic-segmentation SVM model is proposed for multiclass classification. Based on the data set, a mixed integer programming formulation for the chance constrained conic-segmentation SVM is derived. Kernelization of chance constrained conic-segmentation SVM model is also exploited for nonlinear classification. The geometric interpretation is presented to show how the chance constrained conic-segmentation SVM works on uncertain data. Finally, experimental results are presented to demonstrate the effectiveness of the chance constrained conic-segmentation SVM for both artificial and real-world data.

In this paper, we derive deterministic inner approximations for single and joint independent or dependent probabilistic constraints based on classical inequalities from probability theory such as the onesided Chebyshev inequality, Bernstein inequality, Chernoff inequality and Hoeffding inequality (see Pinter, 1989). The dependent case has been modelled via copulas. New assumptions under which the bounds based approximations are convex allowing to solve the problem efficiently are derived. When the convexity condition can not hold, an efficient sequential convex approximation approach is further proposed to solve the approximated problem. Piecewise linear and tangent approximations are also provided for Chernoff and Hoeffding inequalities allowing to reduce the computational complexity of the associated optimization problem. Extensive numerical results on a blend planning problem under uncertainty are finally provided allowing to compare the proposed bounds with the Second Order Cone (SOCP) formulation and Sample Average Approximation (SAA).

In this paper, we consider an n-player non-cooperative game where the random payoff function of each player is defined by its expected value and her strategy set is defined by a joint chance constraint. The random constraint vectors are independent. We consider the case when the probability distribution of each random constraint vector belongs to a subset of elliptical distributions as well as the case when it is a finite mixture of the probability distributions from the subset. We propose a convex reformulation of the joint chance constraint of each player and derive the bounds for players’ confidence levels and the weights used in the mixture distributions. Under mild conditions on the players’ payoff functions, we show that there exists a Nash equilibrium of the game when the players’ confidence levels and the weights used in the mixture distributions are within the derived bounds. As an application of these games, we consider the competition between two investment firms on the same set of portfolios. We use a best response algorithm to compute the Nash equilibria of the randomly generated games of different sizes.

This paper studies an n-player non-cooperative game where each player has expected-value payoff function and chance-constrained strategy set. We consider the case where the row vectors defining the constraints are independent random vectors whose probability distributions are not completely known and belong to a certain distributional uncertainty set. The chance-constrained strategy sets are defined using a distributionally robust framework. We consider one density based uncertainty set and four two-moments based uncertainty sets. One of the considered uncertainty sets is based on a nonnegative support. Under the standard assumptions on the players’ payoff functions, we show that there exists a Nash equilibrium of a distributionally robust chance-constrained game for each uncertainty set. As an application, we study Cournot competition in electricity market and perform the numerical experiments for the case of two electricity firms.

In this paper, a class of stochastic mathematical programs with probabilistic complementarity constraints is considered. We first investigate convergence properties of sample average approximation (SAA) approach to the corresponding chance constrained relaxed complementarity problem. Our discussion can be not only applied to the specific model in this paper, but also viewed as a supplementary for the SAA approach to general joint chance constrained problems. Furthermore, considering the uncertainty of the underlying probability distribution, a distributionally robust counterpart with a moment ambiguity set is proposed. The numerically tractable reformulation is derived. Finally, we use a production planing model to report some preliminary numerical results.