Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems

This paper considers a class of two-stage stochastic mixed-integer optimization problems where, for a given first-stage solution, we can determine the optimal values of recourse variables sequentially. This class of problems arises in a wide variety of applications. In the case of multivariate discrete distributions for uncertain parameters, a standard stochastic programming formulation of these problems involves an exponential number of scenarios, therefore an exponential number of variables and constraints. We propose a new mixed-integer programming modeling approach where the number of variables and constraints is independent of the number of scenarios and scales at most pseudopolynomially with the problem size. The proposed modeling approach relies on state variables that track the system’s state as the uncertainty realizes sequentially. We demonstrate the advantages of the proposed approach in two applications arising in project scheduling and operating room allocation. Summary of Contribution: This paper proposes a new modeling approach for a class of two-stage stochastic optimization problems that is computationally more efficient than the traditional scenario-based stochastic integer programming models. The proposed modeling approach relies on state variables that track the system's state as the uncertainty realizes sequentially. We demonstrated the efficiency of the proposed approach by computational results on two applications in project scheduling and operating room allocation.

We consider combinatorial optimization problems with nonlinear objective functions. Solution approaches for this class of problems proposed so far are either highly problem-specific or they apply generic algorithms for constrained nonlinear optimization, which often does not yield satisfactory results in practice. Our aim is to develop, implement and experimentally evaluate exact algorithms that address the nonlinearity of the objective function and at the same time exploit the underlying combinatorial structure of the problem. To this end we follow two approaches. The first combines good polyhedral descriptions of the objective function and the feasible set in a branch and cut-algorithm. The second approach is based on Lagrangean decomposition. By decomposing the original problem into an unconstrained nonlinear problem and a linear combinatorial problem, we are able to compute strong dual bounds for the optimal value. The computation of lower bounds is then embedded into a branch and bound-algorithm. For many applications there already exist efficient algorithms for the combinatorial subproblem, thus an important aspect of this thesis is the study of the corresponding unconstrained nonlinear subproblems. Both approaches have the advantage that they can easily be adapted to a wide range of nonlinear combinatorial problems.We devise both polyhedral and decomposition- based algorithms for submodular applications from wireless network design and portfolio optimization and evaluate their performance experimentally. Exploiting the equivalence between unconstrained binary quadratic optimization and the maximum cut problem gives rise to a branch and cut-algorithm for quadratic combinatorial problems which we use to compute optimal layouts of tanglegrams, an application from computational biology. Additionally we study the effect of quadratic reformulation of linear constraints, both theoretically and experimentally. The last class of nonlinear combinatorial problems we consider are two-scenario problems. Here we propose a new technique to compute lower bounds in the unconstrained subproblem of the decomposition. Our computational study of the two-scenario minimum spanning tree problem shows that the new Lagrangean decomposition-based algorithm is able to solve significantly larger instances than the standard linearization approach.

The stochastic optimal control solution to the Linear-Quadratic-Gaussian problem with partial information is considered where the system parameters operating on the control are assumed to be unknown Gaussian random processes. This class of problems has special properties that are exploited. First, the state and the random parameters are conditionally Gaussian with respect to the measurement history. This allows the optimal stochastic control problem to be reformulated as a stochastic optimal control problem of full information where the new state space is the state and parameter estimates and the error variance of the state and parameters. Even in this somewhat simple form, there is no closed form solution. Therefore, it is assumed that there is only a first order dependence on the random fluctuation of the parameter about its mean. In this way an asymptotic series solution is obtained in terms of the new state space. For this problem a closed form solution for the controller is obtained up to second order. The controller is derived for the scalar case and then extended to the case of vector state with single input and multiple outputs. The characteristics of the new solution is investigated and a scalar numerial example is given to assess the performance of this new controller.

In this paper, we consider a class of stochastic inverse optimal value problems, in which the forward problem is a linear programming problem (LP), and the data in its constraints are affected by a random variable.The corresponding inverse optimal value problem can be reformulated as a mathematical program with stochastic linear complementarity constraints (MPSLCC).By employing the techniques of sample average approximation (SAA), we construct a series of smooth SAA subproblems and transform them into nonlinear programming problems by utilizing the smooth Fischer-Burmeister function for linear complementarity constraints.In addition, we prove that the sequence of global minimizer (KKT point) of these SAA subproblems converges with probability one (w.p.1) to a global minimizer (an S-stationary point) of MPSLCC under mild conditions.Finally, some numerical experiments are presented to show the availability of our method for solving the given stochastic inverse optimal value problems.

A Class of Linear Quadratic Gaussian Hybrid Optimal Control Problems with Realization–Independent Riccati Equations

This paper introduces a novel class of tree comparison problems strongly motivated by an important and cost intensive step in drug discovery pipeline viz., mapping cell bound receptors to the ligands they bind to and vice versa. Tree comparison studies motivated by problems such as virus-host tree comparison, gene-species tree comparison and consensus tree problem have been reported. None of these studies are applicable in our context because in all these problems, there is a well-defined mapping of the nodes the trees are built on across the set of trees being compared. A new class of tree comparison problems arises in cases where finding the correspondence among the nodes of the trees being compared is itself the problem. The problem arises while trying to find the interclass correspondence between the members of a pair of coevolving classes, e.g., cell bound receptors and their ligands. Given the evolution of the two classes, the combinatorial problem is to find a mapping among the leaves of the two trees that optimizes a given cost function. In this work we formulate various combinatorial optimization problems motivated by the aforementioned biological problem for the first time. We present hardness results, give an efficient algorithm for a restriction of the problem and demonstrate its applicability.

In this paper, we investigate the problem of maximizing the expected exponential utility for an insurer. In the problem setting, the insurer can invest his/her wealth into the market and he/she can also purchase the proportional reinsurance. To control the risk exposure, we impose a value-at-risk constraint on the portfolio, which results in a constrained stochastic optimal control problem. It is difficult to solve a constrained stochastic optimal control problem by using traditional dynamic programming or Martingale approach. However, for the frequently used exponential utility function, we show that the problem can be simplified significantly using a decomposition approach. The problem is reduced to a deterministic constrained optimal control problem, and then to a finite dimensional optimization problem. To show the effectiveness of the approach proposed, we consider both complete and incomplete markets; the latter arises when the number of risky assets are fewer than the dimension of uncertainty. We also conduct numerical experiments to demonstrate the effect of the risk constraint on the optimal strategy.

The stochastic reach-avoid problem and set characterization for diffusions

In this paper, we study a class of stochastic optimal control problems, where the drift term of the equation has a linear growth on the control variable, the cost functional has a quadratic growth, and the control process takes values in a closed set (not necessarily compact). This problem is related to some backward stochastic differential equations (BSDEs) with quadratic growth and unbounded terminal value. We prove that the optimal feedback control exists, and the optimal cost is given by the initial value of the solution of the related BSDE.

In this paper we examine a class of stochastic optimization problems characterized by nondifferentiability of the objective function. It is shown that in many cases the expected value of the objective function is differentiable and thus the resulting optimization problem can be analyzed and solved by using classical analytical or numerical methods. The results are subsequently applied to the solution of a class of stochastic programming problems.

Contributors

Mathematical Analysis of caching policies and cooperation in YouTube-like services

Blocking and Antiblocking Polyhedra

We prove results on bounded solutions to backward stochastic equations driven by random measures. Those bounded BSDE solutions are then applied to solve different stochastic optimization problems with exponential utility in models where the underlying filtration is noncontinuous. This includes results on portfolio optimization under an additional liability and on dynamic utility indifference valuation and partial hedging in incomplete financial markets which are exposed to risk from unpredictable events. In particular, we characterize the limiting behavior of the utility indifference hedging strategy and of the indifference value process for vanishing risk aversion.

The production or marketing portfolio that is optimal under the assumption of quadratic utility may or may not be optimal under the assumption of exponential utility. In certain cases, the necessary and sufficient condition for an identical solution is that absolute risk aversion coefficients associated with the two utility functions be the same. In other cases, equality of risk aversion coefficients is a sufficient condition only. A comparison is made between use of exponential and quadratic utility in the analysis of a California farmer's marketing problem.