Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints

We will propose a branch and bound algorithm for calculating a globally optimal solution of a portfolio construction/rebalancing problem under concave transaction costs and minimal transaction unit constraints. We will employ the absolute deviation of the rate of return of the portfolio as the measure of risk and solve linear programming subproblems by introducing (piecewise) linear underestimating function for concave transaction cost functions. It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner. Received: July 15, 1999 / Accepted: October 1, 2000?Published online December 15, 2000     

On consistency of stochastic dominance and mean–semideviation models

We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. New necessary conditions for stochastic dominance are developed. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, maximization of such a functional yields solutions which are efficient in terms of stochastic dominance. The results are illustrated graphically. Received: September 15, 1998 / Accepted: October 1, 2000?Published online December 15, 2000     

On the core of ordered submodular cost games

A general ordertheoretic linear programming model for the study of matroid-type greedy algorithms is introduced. The primal restrictions are given by so-called weakly increasing submodular functions on antichains. The LP-dual is solved by a Monge-type greedy algorithm. The model offers a direct combinatorial explanation for many integrality results in discrete optimization. In particular, the submodular intersection theorem of Edmonds and Giles is seen to extend to the case with a rooted forest as underlying structure. The core of associated polyhedra is introduced and applications to the existence of the core in cooperative game theory are discussed. Received: November 2, 1995 / Accepted: September 15, 1999?Published online February 23, 2000     

On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating

In previous work, the authors provided a foundation for the theory of variable metric proximal point algorithms in Hilbert space. In that work conditions are developed for global, linear, and super–linear convergence. This paper focuses attention on two matrix secant updating strategies for the finite dimensional case. These are the Broyden and BFGS updates. The BFGS update is considered for application in the symmetric case, e.g., convex programming applications, while the Broyden update can be applied to general monotone operators. Subject to the linear convergence of the iterates and a quadratic growth condition on the inverse of the operator at the solution, super–linear convergence of the iterates is established for both updates. These results are applied to show that the Chen–Fukushima variable metric proximal point algorithm is super–linearly convergent when implemented with the BFGS update. Received: September 12, 1996 / Accepted: January 7, 2000?Published online March 15, 2000     

Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods

The paper extends prior work by the authors on loqo, an interior point algorithm for nonconvex nonlinear programming. The specific topics covered include primal versus dual orderings and higher order methods, which attempt to use each factorization of the Hessian matrix more than once to improve computational efficiency. Results show that unlike linear and convex quadratic programming, higher order corrections to the central trajectory are not useful for nonconvex nonlinear programming, but that a variant of Mehrotra’s predictor-corrector algorithm can definitely improve performance. Received: May 3, 1999 / Accepted: January 24, 2000?Published online March 15, 2000     

Duality and martingales: a stochastic programming perspective on contingent claims

The hedging of contingent claims in the discrete time, discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program. Application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics, namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale. The model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims. However, we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options. An extension of the model to incorporate pre-existing liabilities and endowments reveals the reasons why buyers and sellers trade in options. The model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets. Received: June 5, 2000 / Accepted: July 12, 2001?Published online December 6, 2001     

Optimality conditions for nonconvex semidefinite programming

This paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made. We derive first- and second-order optimality conditions analogous to those for nonlinear programming. Using techniques similar to those used in nonlinear programming, we extend existing theory to cover situations where the constraint matrix is structurally sparse. The discussion covers the case when strict complementarity does not hold. The regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal. Received: May 15, 1998 / Accepted: April 12, 2000?Published online May 12, 2000     

Error bounds for 2-regular mappings with Lipschitzian derivatives and their applications

We obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature. Specifically, we assume that the constraints mapping has a Lipschitzian derivative, and satisfies a certain 2-regularity condition at the point under consideration. This setting directly subsumes the classical regular case and the twice differentiable 2-regular case, for which error bounds are known, but it is significantly richer than either of these two cases. When applied to a certain equation-based reformulation of the nonlinear complementarity problem, our results yield an error bound under an assumption more general than b-regularity. The latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available. We also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001     

Existence of optimal solutions and duality results under weak conditions

In this paper we consider an ordinary convex program with no qualification conditions (such as Slater’s condition) and for which the optimal set is neither required to be compact, nor to be equal to the sum of a compact set and a linear space. It is supposed only that the infimum α is finite. A very wide class of convex functions is exhibited for which the optimum is always attained and α is equal to the supremum of the ordinary dual program. Additional results concerning the existence of optimal solutions in the non convex case are also given. Received: February 1, 1999 / Accepted: December 15, 1999?Published online February 23, 2000