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This paper deals with a problem of guaranteed (robust) financial decision-making under model uncertainty. An efficient method is proposed for determining optimal robust portfolios of risky financial instruments in the presence of ambiguity (uncertainty) on the probabilistic model of the returns. Specifically, it is assumed that a nominal discrete return distribution is given, while the true distribution is only known to lie within a distance d from the nominal one, where the distance is measured according to the Kullback-Leibler divergence. The goal in this setting is to compute portfolios that are worst-case optimal in the mean-risk sense, that is, to determine portfolios that minimize the maximum with respect to all the allowable distributions of a weighted risk-mean objective. The analysis in the paper considers both the standard variance measure of risk and the absolute deviation measure. [ABSTRACT FROM AUTHOR]

In this paper we use the penalty approach to study three constrained minimization problems. A penalty function is said to have the exact penalty property [J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, 2 vols., Springer-Verlag, Berlin, 1993] if there exists a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish a very simple sufficient condition for the exact penalty property. [ABSTRACT FROM AUTHOR]

This paper presents a new verified optimization method for the problem of finding the densest packings of nonoverlapping equal circles in a square. In order to provide reliable numerical results, the developed algorithm is based on interval analysis. As one of the most efficient parts of the algorithm, an interval-based version of a previous elimination procedure is introduced. This method represents the remaining areas still of interest as polygons fully calculated in a reliable way. Currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems. Still, as a result of the highly increasing number of subproblems, earlier computer-aided methods were not able to solve problem instances where the number of circles was greater than 27. The present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together. As a demonstration of the capabilities of the new algorithm the problems of packing 28, 29, and 30 circles were solved within very tight tolerance values. Our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of person-by-person (pbp) optimization to binary decision spaces. The novelty of our approach is the adaptation to a dynamic team context of notions borrowed from the pseudo-boolean optimization field as completely local-global or unimodal functions and submodularity. We also generalize the concept of pbp optimization to the case where groups of m decisions makers make joint decisions sequentially, which we refer to as mbm optimization. The main contribution is a description of sufficient conditions, verifiable in polynomial time, under which a pbp or an mbm optimization algorithm converges to the team-optimum. As a second contribution, we present a local and greedy algorithm characterized by approximate decision strategies (i.e., strategies based on a local state vector) that return the same decisions as in the complete information framework (where strategies are based on full state vector). As a last contribution, we also show that there exists a subclass of submodular team problems, recognizable in polynomial time, for which the pbp optimization converges for at least an opportune initialization of the algorithm. [ABSTRACT FROM AUTHOR]

The paper proposes a general theory for the treatment of a class of optimization problems that arise as a consequence of a search for the optimal replenishment schedule for finite time horizon inventory models. The decision variables in these optimization problems consist of the number of replenishment periods and the lengths of the periods. When the number of replenishment periods is fixed, it is shown that the optimization problems have a unique solution under some partial differential inequalities. Furthermore, the minimal value is convex as a function of the number of replenishment periods. This leads to a simple procedure for determining the optimal control policy. [ABSTRACT FROM AUTHOR]

The dogleg method is a classical trust-region technique for globalizing Newton's method. While it is widely used in optimization, including large-scale optimization via truncated- Newton approaches, its implementation in general inexact Newton methods for systems of nonlinear equations can be problematic. In this paper, we first outline a very general dogleg method suitable for the general inexact Newton context and provide a global convergence analysis for it. We then discuss certain issues that may arise with the standard dogleg implementational strategy and propose modified strategies that address them. Newton-Krylov methods have provided important motivation for this work, and we conclude with a report on numerical experiments involving a Newton-GMRES dogleg method applied to benchmark CFD problems. [ABSTRACT FROM AUTHOR]

The goal of this paper is to give a rigorous justification for the Hessian-constrained problems introduced in [G. Bouchitté and I. Fragal`a, Arch. Ration. Mech. Anal., 184 (2007), pp. 257-284] and to show how they are linked to the optimal design of thin plates. To that aim, we study the asymptotic behavior of a sequence of optimal elastic compliance problems in the double limit when both the maximal height of the design region and the total volume of the material tend to zero. In the vanishing volume limit, a sequence of linear constrained first order vector problems is obtained, which in turn-in the vanishing thickness limit-produces a new linear constrained problem where both first and second order gradients appear. When the load is orthogonal to the plate, only the Hessian constraint is active, and we recover as a particular case the optimization problem studied in [G. Bouchitté and I. Fragal`a, Arch. Ration. Mech. Anal., 184 (2007), pp. 257-284] (see also [T. Lewinski and J. J. Telega, Arch. Mech. (Arch. Mech. Stos.), 53 (2001), pp. 457-485]). [ABSTRACT FROM AUTHOR]

In this paper we study the regularity properties of a free boundary problem arising in the optimization of the best Sobolev trace constant in the immersion $H^1(\Omega)\hookrightarrow L^q(\partial\Omega)$ for functions that vanish in a subset of $\Omega$. This problem is also related to a minimization problem for Steklov eigenvalues. [ABSTRACT FROM AUTHOR]

We introduce a new barrier-type function which is not, a barrier function in the usual sense: it has finite value at the boundary of the feasible region. Despite this, the iteration bound of a large-update interior-point method based on this function is shown to be O(square root of n (log n) log n/epsilon), which is as good as tile currently best known bound for large update methods. The recently introduced properly of exponential convexity for the kernel function underlying the barrier function, as well as the strong convexity of the kernel function, are crucial in the analysis. [ABSTRACT FROM AUTHOR]

This work deals with bound constrained multiobjective optimization (MOP) of nonsmooth functions for problems where the structure of the objective functions either cannot be exploited, or are absent. Typical situations arise when the functions are computed as the result of a computer simulation. We first present definitions and optimality conditions as well as two families of single-objective formulations of MOP. Next, we propose a new algorithm called BiMads for the biobjective optimization (BOP) problem (i.e., MOP with two objective functions). The property that Pareto points may be ordered in BOP and not in MOP is exploited by our algorithm. BiMads generates an approximation of the Pareto front by solving a series of single-objective formulations of BOP. These single-objective problems are solved using the recent Mads (mesh adaptive direct search) algorithm for nonsmooth optimization. The Pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the Clarke calculus. Finally, BiMads is tested on problems from the literature designed to illustrate specific difficulties encountered in biobjective optimization, such as a nonconvex or disjoint Pareto front, local Pareto fronts, or a nonuniform Pareto front. [ABSTRACT FROM AUTHOR]

A new nonlinear conjugate gradient method and an associated implementation, based on an inexact line search, are proposed and analyzed. With exact line search, our method reduces to a nonlinear version of the Hestenes--Stiefel conjugate gradient scheme. For any (inexact) line search, our scheme satisfies the descent condition ${\bf g}_k^{\sf T} {\bf d}_k \le -\frac{7}{8} \|{\bf g}_k\|^2$. Moreover, a global convergence result is established when the line search fulfills the Wolfe conditions. A new line search scheme is developed that is efficient and highly accurate. Efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer. High accuracy is achieved by using a convergence criterion, which we call the ``approximate Wolfe'' conditions, obtained by replacing the sufficient decrease criterion in the Wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion. Numerical comparisons are given with both L-BFGS and conjugate gradient methods using the unconstrained optimization problems in the CUTE library. [ABSTRACT FROM AUTHOR]

We discuss a multigrid approach to the optimization of systems governed by differential equations. Such optimization problems appear in many applications and are of a different nature than systems of equations. Our approach uses an optimization-based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids. Our goal is not to argue for a particular optimization-based multigrid algorithm, but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms. Furthermore, using several model problems we give evidence (both theoretical and numerical) that the optimization setting is well suited to multigrid algorithms. Some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation. In addition, we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization-based multigrid algorithm. [ABSTRACT FROM AUTHOR]