Compact Constraint Set Research Articles

Singular optimal control by minimizer flows

This paper studies a computational method to deal with a singular optimal control problem by minimizer flows in a viscosity approximation to the Hamilton–Jacobi–Bellman equation. The boundary of the compact constraint set of control variable is intersected with a class of minimizer flows to yield a Hamiltonian extremal function in rewriting the HJB equation. The analysis properties of the flow are revealed in a global optimization framework. An example on computing a minimizer flow and a Hamiltonian extremal function is presented. An application of the minimizer flow in a non-smooth optimization problem is also mentioned.

Stability of semi-infinite vector optimization problems without compact constraints

This paper is devoted to the continuity of solution maps for perturbation semi-infinite vector optimization problems without compact constraint sets. The sufficient conditions for lower semicontinuity and upper semicontinuity of solution maps under functional perturbations of both objective functions and constraint sets are established. Some examples are given to analyze the assumptions in the main result.

A 2-approximation for minmax regret problems via a mid-point scenario optimal solution

In this note, a 2-approximation method for minmax regret optimization problems is developed which extends the work of Kasperski and Zielinski [A. Kasperski, P. Zielinski, An approximation algorithm for interval data minmax regret combinatorial optimization problems, Information Processing Letters 97 (2006) 177–180] from finite to compact constraint sets.

The core of social choice problems with monotone preferences and a feasibility constraint

We consider collective choice with agents possessing strictly monotone, strictly convex and continuous preferences over a compact and convex constraint set contained in +k . If it is non-empty the core will lie on the efficient boundary of the constraint set and any policy not in the core is beaten by some policy on the efficient boundary. It is possible to translate the collective choice problem on this efficient boundary to another social choice problem on a compact and convex subset of +c (c<k) with strictly convex and continuous preferences. In this setting the dimensionality results in Banks (1995) and Saari (1997) apply to the dimensionality of the boundary of the constraint set (which is lower than the dimensionality of the choice space by at least one). If the constraint set is not convex then the translated lower dimensional problem does not necessarily involve strict convexity of preferences but the dimensionality of the problem is still lower. Broadly, the results show that the homogeneity afforded by strict monotonicity of preferences and a compact constraint set makes generic core emptyness slightly less common. One example of the results is that if preferences are strictly monotone and convex on 2 then choice on a compact and convex constraint exhibits a version of the median voter theorem.

Policy iteration type algorithms for recurrent state Markov decision processes

We introduce and analyze several new policy iteration type algorithms for average cost Markov decision processes (MDPs). We limit attention to “recurrent state” processes where there exists a state which is recurrent under all stationary policies, and our analysis applies to finite-state problems with compact constraint sets, continuous transition probability functions, and lower-semicontinuous cost functions. The analysis makes use of an underlying relationship between recurrent state MDPs and the so-called stochastic shortest path problems of Bertsekas and Tsitsiklis (Math. Oper. Res. 16(3) (1991) 580). After extending this relationship, we establish the convergence of the new policy iteration type algorithms either to optimality or to within ε>0 of the optimal average cost.

On terminating Markov decision processes with a risk-averse objective function

We consider a class of terminating Markov decision processes with an exponential risk-averse objective function and compact constraint sets. We assume the existence of an absorbing cost-free terminal state Ω, positive transition costs and continuity of the transition probability and cost functions. Without discounting future costs in the argument of the exponential utility function, we establish (i) the existence of a real-valued optimal cost function which can be achieved by a stationary policy and (ii) the convergence of value iteration and policy iteration to the unique solution of Bellman's equation. We illustrate the results with two computational examples.

Asymptotic properties of some classes ofM-estimates

The article considers a number of regression models with unknown coefficients. We describe some of these models and prove general propositions concerning their asymptotic behavior. The parametric set of unknown parameters is always assumed closed and in general unbounded. The case of an open set is simpler in our view, and at least the asymptotic distribution of the estimates in this case is generally normal under natural constraints, which is not always so with a compact constraint set. In what follows we consider criteria and estimates of a general form that hmlude many hitherto ignored cases of regression analysis. These are primarily M-estimates in P. Huber's terminology, which constitute the most natural class of robust regression estimates. We consider various classes of M-estimates for parametric and nonparametric regression models, which allow observation errors that are dependent at different time instants. We always assume discrete time; the continuous time case is not considered here, although many of our propositions are also valid for continuous time.