Today, stochastic programming has established itself as a powerful modeling tool when an accurate probabilistic description of the randomness is available; however, in many
real-life applications the decision-maker does not have this information, for instance when it comes to estimating financial stock returns. The need for an alternative, non- probabilistic, theory of decision-making under uncertainty has become pressing in recent years because of volatile market conditions and unstable economical states, which reduce the amount of reliable information available and make it obsolete more quickly.
Traditional models of decision making under uncertainty assume perfect information, i.e. accurate values for the system parameters and specific probability distributions for the random variables. However, such information is rarely available in practice. Soyster addressed this issue in his work [135] in the early 1970s, where every uncertain param- eter in convex programming problems was taken equal to its worst-case value within a set. While this achieved the desired effect of immunizing the problem against parame- ter uncertainty, it was widely considered too conservative for practical implementation. Ben-Tal and Nemirovski [10, 11, 12] and El-Ghaoui and Lebret [45, 46] addressed the issue of over conservatism by restricting the uncertain parameters to belong to ellip- soidal uncertainty sets, which removes the most unlikely outcomes from consideration and yields tractable mathematical programming problems. A drawback of this method is that it increases the complexity of the problem considered, e.g., the robust counter- part of a linear programming problem is a second-order cone problem. More recently, Bertsimas and Sim [20, 21] and Bertsimas et. al. [17] have proposed a robust opti- mization approach based on polyhedral uncertainty sets, which preserves the class of problems under analysis, e.g., the robust counterpart of a linear programming problem remains a linear programming problem, and thus has advantages in terms of tractability in large-scale settings. It can also be connected to the decision maker’s attitude towards uncertainty, providing guidelines to construct the uncertainty set from the historical realizations of the random variables using data-driven optimization [15].
2.5.1 Problem of moments
Given historical data, it is easier to estimate moment information of random parame- ters than to derive their probability distributions. This motivates the use of moment information in developing uncertainty models for random parameters. The problem of moments and its variations have been extensively studied and applied to many opti- mization problems in the literature.
The problem of moment has been studied by Stieltjes [136] in the ninetheenth cen- tury. The problem is related to the characterization of a feasible sequence of moments. Schmudgen [126], Putinar [110], and Curto and Fialkow [30] derived necessary and suffi- cient conditions sequences of moments with different settings. The problem of moments is also related to optimization over polynomials (the dual theory of moment). Lasserre [87] and Parrilo [107] among others proposed relaxation hierarchies for optimization over polynomials using moment results.
Bertsimas and Popescu [19] further studied the optimal inequalities given moment in- formation. Moment problems in finance such as option pricing problems have been investigated in the literature (see [18, 94, 23]).
Stochastic Programs with Second
Order Stochastic Dominance
Constraints
3.1
Overview
Inspired by the successful applications of the stochastic optimization with second order stochastic dominance (SSD) model in portfolio optimization, we study new numerical methods for a general SSD model where the underlying functions are not necessarily linear. Specifically, we penalize the SSD constraints to the objective and then apply the well known stochastic approximation (SA) method and the level function methods to solve the penalized problem. Both methods are iterative: the former requires the calculation of only one approximate subgradient per iteration and can be applied to the case when the underlying functions are highly nonlinear and/or non-smooth, and the distribution of the random variable may be unknown.
The main contribution of this chapter can be summarized as follows:
• We exploit a recently developed exact penalization scheme for stochastic program- ming models with SSD constraints and apply the stochastic approximation method and the level function methods to solve the penalized problem.
• We apply the penalization scheme and the numerical methods to some portfolio problems where the underlying return functions are not necessarily linear and present some test results. Moreover, we use real world test data to set up both backtest and out-of-sample test for investigating the performance of the portfolio based on the SSD model in comparison with the Markowitz model. Furthermore, a nonlinear supply chain problem is introduced, and the performance of the level
function methods along with the cutting plane method discussed in [81, 52] is investigated.
Throughout this chapter, we use the following notation. Let xTy denotes the scalar products of two vectors x and y, and let k · k denotes the Euclidean norm. For a real valued smooth function h(x), we use ∇h(x) to denote the gradient of h at x. Let “conv” denotes the convex hull of a set.
The rest of this chapter is organized as follows. In Section 3.2, we introduce the opti- mization problem and discuss preliminaries needed throughout the chapter. In Section 3.3 we discuss the stochastic quasi-gradient algorithm and the level function algorithms and analyze the convergence of optimal solutions. In Section 3.4, we apply the proposed methods to portfolio optimization problems, a supply chain problem and report some numerical test results. Finally, in Section 3.5 we present some conclusions.