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For problems in the calculus of variations with isoperimetric side constraints, we provide in this paper a set of points whose emptiness, independently of nonsingularity assumptions, is equivalent to the nonnegativity of the second variation along admissible variations. The main objective of introducing a characterization of this condition should be, of course, to obtain a simpler way of verifying it. There are two other sets of points available in the literature, introduced by Loewen and Zheng (1994) and Zeidan (1996), for which this necessary condition implies their emptiness. However, we show that verifying membership of these sets may be more difficult than checking directly if that condition holds. Contrary to this behavior, we prove that the desired objective of characterizing that condition is achieved by means of the set introduced in this paper. [ABSTRACT FROM AUTHOR]

In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional $\mathcal {J}(u)=\int_{\partial\Omega}f(x)u\,\mathrm {d}\mathcal {H}^{N-1}$ over some admissible class of loads f where u is the (unique) solution to the problem −Δ p u+| u| p−2 u=0 in Ω with | ∇ u| p−2 u ν= f on ∂Ω. [ABSTRACT FROM AUTHOR]

We consider a problem of finding optimal contracts in continuous time, when the agent’s actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal’s utility is infinite, while it becomes finite with hidden action, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization. [ABSTRACT FROM AUTHOR]

This paper concerns discrete-time multiobjective Markov control processes on Borel spaces and unbounded costs. Under mild assumptions, it is shown that the usual “scalarization approach” to obtain Pareto policies for the multiobjective control problem is in fact equivalent to solving the dual of a certain multiobjective infinite-dimensional linear program. The latter program is obtained from a multiobjective measure problem which is also used to prove the existence of strong Pareto policies, that is, Pareto policies whose cost vector is the closest to the control problem’s virtual minimum. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study optimal control of generalized quasi-variational hemivariational inequalities involving multivalued mapping. Under some suitable conditions, we give existence results of the optimal control. We also consider the convergence behavior of the optimal control when the data for the underlying quasi-variational hemivariational inequalities is contaminated by some noise. In the last section, we give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper focuses on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. For rectangular domains with Neumann boundary condition, it is known that there exists a threshold value such that if the total weight is below this threshold value then the optimal favorable region is like a section of a disk at one of the four corners; otherwise, the optimal favorable region is a strip attached to the shorter side of the rectangle. Here, we investigate the same problem with mixed Robin-Neumann type boundary conditions and study how this boundary condition affects the optimal spatial arrangement. [ABSTRACT FROM AUTHOR]

In this paper a general model of a market with asset prices and economical factors of Markovian structure is considered. The problem is to find optimal portfolio strategies maximizing a discounted infinite horizon reward functional consisting of an integral term measuring the quality of the portfolio at each moment and a discrete term measuring the reward from consumption. There are general transaction costs which, in particular, cover fixed plus proportional costs. It is shown, under general conditions, that there exists an optimal impulse strategy and the value function is a solution to the Bellman equation which corresponds to suitable quasi-variational inequalities. [ABSTRACT FROM AUTHOR]

The order reduction method for singularly perturbed optimal control systems consists of setting the small parameter equal to zero and employing the differential system thus obtained. Although in many situations this provides the correct variational limit problem, it is established in this paper that when considering systems with non-scalar fast variables, the set of systems for which the order reduction method is invalid is dense in the class of systems under consideration. This extends previous results, where only systems with linear fast variables were considered. The present result complements a result established in a joint work with Artstein, where it was established that the order reduction method is valid for singularly perturbed optimal control systems with scalar fast variable. [ABSTRACT FROM AUTHOR]

The background for this article is the question of modification of the geometric configuration of an elastic structure by means of "volume"-type actuation. In this actuation mode stresses are applied to the elastic body by injection/extraction of a fluid into, or from, a large number of vacuoles in the elastic "matrix" material. Previous articles by the author, and others, have examined this process and studied its effectiveness in the context of a "naive" continuous model. The present paper develops modified models in the two-dimensional case by applying the "multiple scales" homogenization method to elastic bodies with increasingly small and numerous vacuoles arranged in a rectangularly periodic structure. The process leads to a modification of the naive model potentially useful in future elastic system control research. [ABSTRACT FROM AUTHOR]

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda>0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p<+\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings. [ABSTRACT FROM AUTHOR]

In this paper we study the asset pricing problem when the volatility is random. First, we derive a PDE for the risk-minimizing price of any contingent claim. Secondly, we assume that the volatility process at is observed through an observation process Y[sub t] subject to random error. A price formula and a PDE are then derived regarding the stock price S[sub t] and the observation process Y[sub t] as parameters. Finally, we assume that S[sub t] is observed. In this case we have a complete market and any contingent claim is then priced by an arbitrage argument instead of by risk-minimizing. [ABSTRACT FROM AUTHOR]

A family of parametric linear-quadratic optimal control problems is considered. The problems are subject to state constraints. It is shown that if weak second-order sufficient optimality conditions and standard constraint qualifications are satisfied at the reference point, then, for small perturbations of the parameter, there exists a locally unique stationary point, corresponding to a solution. This point is a Lipschitz continuous function of the parameter. [ABSTRACT FROM AUTHOR]

The problem of finding the mean variance optimal portfolio in a multiperiod model can not be solved directly by means of dynamic programming. In order to find a solution we therefore first introduce independent market clones having the same distributional properties as the original market, and we replace the portfolio mean and variance by their empirical counterparts. We then use dynamic programming to derive portfolios maximizing a weighted sum of the empirical mean and variance. By letting the number of market clones converge to infinity we are able to solve the original mean variance problem. [ABSTRACT FROM AUTHOR]

The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhäuser, Basel, ) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101–117, ). Although, the problem in one dimension has a solution (cf. Kreĭn in AMS Transl. Ser. 2(1):163–187, ) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185–220, ), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328–337, ). [ABSTRACT FROM AUTHOR]

We address a portfolio optimization problem in a semi-Markov modulated market. We study both the terminal expected utility optimization on finite time horizon and the risk-sensitive portfolio optimization on finite and infinite time horizon. We obtain optimal portfolios in relevant cases. A numerical procedure is also developed to compute the optimal expected terminal utility for finite horizon problem. [ABSTRACT FROM AUTHOR]

Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, second-order cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient matrix of the equation which is determined by the sparsity of an optimization problem (linear program, semidefinite program or second-order cone program). We show if an optimization problem is correlatively sparse, then the coefficient matrix of the Schur complement equation inherits the sparsity, and a sparse Cholesky factorization applied to the matrix results in no fill-in. [ABSTRACT FROM AUTHOR]

We consider degenerate parabolic and elliptic fully nonlinear Bellman equations with Lipschitz coefficients in domains. Error bounds of order h1/2 in the sup norm for certain types of finite-difference schemes are obtained. [ABSTRACT FROM AUTHOR]

We make the link between two approaches to Nash equilibria for nonzero-sum stochastic differential games: the first one using backward stochastic differential equations and the second one using strategies with delay. We prove that, when both exist, the two notions of Nash equilibria coincide. [ABSTRACT FROM AUTHOR]

We study an optimization problem for an elliptic PDE where the unknown is a measure. The state equation is a relaxed form of the classical Dirichlet problem and the functional to minimize is nonlinear. We prove a regularity result for the optimal measure, give a description of it and present some numerical examples. [ABSTRACT FROM AUTHOR]

We consider ruin probabilities for an insurance company, which can also invest in the stock market. The risk process is modeled by a compound Poisson process and the stock price by geometric Brownian motion. We show that if the tails of the claims are light tailed, then the optimal strategy is asymptotically given by holding a constant $-value in the stock position. Furthermore, we show that a kind of Cramér–Lundberg approximation holds for the minimal ruin probability. Everything is shown under assumptions, which are analogous to the assumptions in the case of the classical Cramér–Lundberg approximation without investment. [ABSTRACT FROM AUTHOR]