Augmented Lagrangians and Marginal Values in Parameterized Optimization Problems

When an optimization problem depends on parameters, the minimum value in the problem as a function of the parameters is typically far from being differentiable. Certain subderivatives nevertheless exist and can be intepreted as generalized marginal values. In this paper such subderivatives are studied in an abstract setting that allows for infinite dimensionality of the decision space. By means of the notion of proximal subgradients, a new general formula of subdifferentiation is established which provides an upper bound for the marginal values in question and a very broad criterion for local Lipschitz continuity of the optimal value function. Augmented Lagrangians are introduced and shown to lead to still sharper estimates in terms of special multiplier vectors. This approach opens a way to taking higher-order optimality conditions into account in such estimates

Favorable Classes of Lipschitz Continuous Functions in Subgradient Optimization

Clarke has given a robust definition of subgradients of arbitrary Lipschitz continuous functions f on R^n, but for purposes of minimization algorithms it seems essential that the subgradient multifunction partial f have additional properties, such as certain special kinds of semicontinuity, which are not automatic consequences of f being Lipschitz continuous. This paper explores properties of partial f that correspond to f being subdifferentially regular, another concept of Clarke's, and to f being a pointwise supremum of functions that are k times continuously differentiable

Lipschitzian Stability in Optimization: The Role of Nonsmooth Analysis

The motivations of nonsmooth analysis are discussed. Appiications are given to the sensitivity of optimal vaiues, the interpretation of Lagrange multipliers, and the stabiiity of constraint systems under perturbation

Convexity and Duality in Hamilton-Jacobi Theory

Value functions propagated from initial or terminal costs and constraints by way of a differential or more broadly through a Lagrangian that may take on "alpha," are studied in the case where convexity persists in the state argument. Such value functions, themselves taking on "alpha," are shown to satisfy a subgradient form of the Hamilton-Jacobi equation which strongly supports properties of local Lipschitz continuity, semidifferentibility and Clarke regularity. An extended `method of characteristics' is developed which determines them from Hamiltonian dynamics underlying the given Lagrangian. Close relations with a dual value function are revealed

Asymptotic Theory for Solutions in Generalized M-Estimation and Stochastic Programming

New techniques of local sensitivity analysis in nonsmooth optimization are applied to the problem of studying the asymptotic behavior (generally non-normal) for solutions in stochastic optimization, and generalized M-estimation -- a reformulation of the traditional maximum-likelihood problem that allows the introduction of hard constraints

Arbitrage and deflators in illiquid markets

This paper presents a stochastic model for discrete-time trading in financial markets where trading costs are given by convex cost functions and portfolios are constrained by convex sets. The model does not assume the existence of a cash account/numeraire. In addition to classical frictionless markets and markets with transaction costs or bid-ask spreads, our framework covers markets with nonlinear illiquidity effects for large instantaneous trades. In the presence of nonlinearities, the classical notion of arbitrage turns out to have two equally meaningful generalizations, a marginal and a scalable one. We study their relations to state price deflators by analyzing two auxiliary market models describing the local and global behavior of the cost functions and constraints

Generalized Linear-Quadratic Problems of Deterministic and Stochastic Optimal Control in Discrete Time

Two fundamental classes of problems in large-scale linear and quadratic programming are described. Multistage problems covering a wide variety of models in dynamic programming and stochastic programming are represented in a new way. Strong properties of duality are revealed which support the development of iterative approximate techniques of solution in terms of saddlepoints. Optimality conditions are derived in a form that emphasizes the possibilities of decomposition

On the Interchange of Subdifferentiation and Conditional Expectation for Convex Functionals

We show that the operators E^G (conditional expectation given a tau-field G) and partial (subdifferentiation), when applied to a normal convex integrand f, commute if the effective domain multifunction omega -> {x E R^n | f(omega , x) < +infinity } is G-measurable

Deterministic and Stochastic Optimization Problems of Bolza Type in Discrete Time

In this paper we consider deterministic and stochastic versions of discrete time analogs of optimization problems of the Bolza type. The functionals are assumed to be convex, but we make no differentiability assumptions and allow for the explicit or implicit presence of constraints both on the state x_t and the increments delta x_t. The deterministic theory serves to set the stage for the stochastic problem. We obtain optimality conditions that are always sufficient and which are also necessary if the given problem satisfies a strict feasibility condition and, in the stochastic case, a bounded recourse condition. This is a new condition that bypasses the uniform boundedness restrictions encountered in earlier work on related problems

Scenarios and Policy Aggregation in Optimization under Uncertainty

A common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really strong enough to support a stochastic programming model, has been to set up and analyze a number of scenarios. The aim then is to identify trends and essential features on which a robust decision policy can be based. This paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios. The scenarios are bundled at various levels to reflect the availability of information, and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight