Inverse Optimization of Convex Risk Functions

The theory of convex risk functions has now been well established as the basis for identifying the families of risk functions that should be used in risk averse optimization problems. Despite its theoretical appeal, the implementation of a convex risk function remains difficult, as there is little guidance regarding how a convex risk function should be chosen so that it also well represents one's own risk preferences. In this paper, we address this issue through the lens of inverse optimization. Specifically, given solution data from some (forward) risk-averse optimization problems we develop an inverse optimization framework that generates a risk function that renders the solutions optimal for the forward problems. The framework incorporates the well-known properties of convex risk functions, namely, monotonicity, convexity, translation invariance, and law invariance, as the general information about candidate risk functions, and also the feedbacks from individuals, which include an initial estimate of the risk function and pairwise comparisons among random losses, as the more specific information. Our framework is particularly novel in that unlike classical inverse optimization, no parametric assumption is made about the risk function, i.e. it is non-parametric. We show how the resulting inverse optimization problems can be reformulated as convex programs and are polynomially solvable if the corresponding forward problems are polynomially solvable. We illustrate the imputed risk functions in a portfolio selection problem and demonstrate their practical value using real-life data

Computable optimal value bounds for generalized convex programs

It has been shown by Fiacco that convexity or concavity of the optimal value of a parametric nonlinear programming problem can readily be exploited to calculate global parametric upper and lower bounds on the optimal value function. The approach is attractive because it involves manipulation of information normally required to characterize solution optimality. A procedure is briefly described for calculating and improving the bounds as well as its extensions to generalized convex and concave functions. Several areas of applications are also indicated

First-order methods of smooth convex optimization with inexact oracle

In this paper, we analyze different first-order methods of smooth convex optimization employing inexact first-order information. We introduce the notion of an approximate first-order oracle. The list of examples of such an oracle includes smoothing technique, Moreau-Yosida regularization, Modified Lagrangians, and many others. For different methods, we derive complexity estimates and study the dependence of the desired accuracy in the objective function and the accuracy of the oracle. It appears that in inexact case, the superiority of the fast gradient methods over the classical ones is not anymore absolute. Contrary to the simple gradient schemes, fast gradient methods necessarily suffer from accumulation of errors. Thus, the choice of the method depends both on desired accuracy and accuracy of the oracle. We present applications of our results to smooth convex-concave saddle point problems, to the analysis of Modified Lagrangians, to the prox-method, and some others.smooth convex optimization, first-order methods, inexact oracle, gradient methods, fast gradient methods, complexity bounds

Constructing a subgradient from directional derivatives for functions of two variables

For any scalar-valued bivariate function that is locally Lipschitz continuous and directionally differentiable, it is shown that a subgradient may always be constructed from the function's directional derivatives in the four compass directions, arranged in a so-called "compass difference". When the original function is nonconvex, the obtained subgradient is an element of Clarke's generalized gradient, but the result appears to be novel even for convex functions. The function is not required to be represented in any particular form, and no further assumptions are required, though the result is strengthened when the function is additionally L-smooth in the sense of Nesterov. For certain optimal-value functions and certain parametric solutions of differential equation systems, these new results appear to provide the only known way to compute a subgradient. These results also imply that centered finite differences will converge to a subgradient for bivariate nonsmooth functions. As a dual result, we find that any compact convex set in two dimensions contains the midpoint of its interval hull. Examples are included for illustration, and it is demonstrated that these results do not extend directly to functions of more than two variables or sets in higher dimensions.Comment: 16 pages, 2 figure