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## Numerical Calibration of Steiner trees

### Applied Mathematics & Optimization (2017-05-10): 1-18 , May 10, 2017

In this paper we propose a variational approach to the Steiner tree problem, which is based on calibrations in a suitable algebraic environment for polyhedral chains which represent our candidates. This approach turns out to be very efficient from numerical point of view and allows to establish whether a given Steiner tree is optimal. Several examples are provided.

## Stochastic Maximum Principle for Optimal Control of SPDEs

### Applied Mathematics & Optimization (2013-10-01) 68: 181-217 , October 01, 2013

We prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a stochastic partial differential equation driven by a finite dimensional Wiener process. The equation is formulated in a semi-abstract form that allows direct applications to a large class of controlled stochastic parabolic equations. We allow for a diffusion coefficient dependent on the control parameter, and the space of control actions is general, so that in particular we need to introduce two adjoint processes. The second adjoint process takes values in a suitable space of operators on *L*^{4}.

## A Semi-linear Backward Parabolic Cauchy Problem with Unbounded Coefficients of Hamilton–Jacobi–Bellman Type and Applications to Optimal Control

### Applied Mathematics & Optimization (2015-08-01) 72: 1-36 , August 01, 2015

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.

## Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

### Applied Mathematics & Optimization (2003-03-01) 47: 121-142 , March 01, 2003

*Abstract. * Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.

## Optimal Consumption in a Brownian Model with Absorption and Finite Time Horizon

### Applied Mathematics & Optimization (2013-04-01) 67: 197-241 , April 01, 2013

We construct *ϵ*-optimal strategies for the following control problem: Maximize
$\mathbb {E}[ \int_{[0,\tau)}e^{-\beta s}\,dC_{s}+e^{-\beta\tau}X_{\tau}]$
, where *X*_{t}=*x*+*μt*+*σW*_{t}−*C*_{t}, *τ*≡inf{*t*>0|*X*_{t}=0}∧*T*, *T*>0 is a fixed finite time horizon, *W*_{t} is standard Brownian motion, *μ*, *σ* are constants, and *C*_{t} describes accumulated consumption until time *t*. It is shown that *ϵ*-optimal strategies are given by barrier strategies with time-dependent barriers.

## Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms

### Applied Mathematics & Optimization (2010-12-01) 62: 411-434 , December 01, 2010

While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: *When is the product of finitely many positive definite quadratic forms convex, and what is the Legendre-Fenchel transform for it*? First, we show that the convexity of the product is determined intrinsically by the condition number of so-called ‘scaled matrices’ associated with quadratic forms involved. The main result claims that if the condition number of these scaled matrices are bounded above by an explicit constant (which depends only on the number of quadratic forms involved), then the product function is convex. Second, we prove that the Legendre-Fenchel transform for the product of positive definite quadratic forms can be expressed, and the computation of the transform amounts to finding the solution to a system of equations (or equally, finding a Brouwer’s fixed point of a mapping) with a special structure. Thus, a broader question than the open “Question 11” in Hiriart-Urruty (SIAM Rev. 49, 225–273, 2007) is addressed in this paper.

## On a Class of Nonlinear Viscoelastic Kirchhoff Plates: Well-Posedness and General Decay Rates

### Applied Mathematics & Optimization (2016-02-01) 73: 165-194 , February 01, 2016

This paper is concerned with well-posedness and energy decay rates to a class of nonlinear viscoelastic Kirchhoff plates. The problem corresponds to a class of fourth order viscoelastic equations of $$p$$ -Laplacian type which is not locally Lipschitz. The only damping effect is given by the memory component. We show that no additional damping is needed to obtain uniqueness in the presence of rotational forces. Then, we show that the general rates of energy decay are similar to ones given by the memory kernel, but generally not with the same speed, mainly when we consider the nonlinear problem with large initial data.

## Stochastic Control of Memory Mean-Field Processes

### Applied Mathematics & Optimization (2017-06-17): 1-24 , June 17, 2017

By a memory mean-field process we mean the solution
$$X(\cdot )$$
of a stochastic mean-field equation involving not just the current state *X*(*t*) and its law
$$\mathcal {L}(X(t))$$
at time *t*, but also the state values *X*(*s*) and its law
$$\mathcal {L}(X(s))$$
at some previous times
$$s<t.$$
Our purpose is to study stochastic control problems of memory mean-field processes. We consider the space
$$\mathcal {M}$$
of measures on
$$\mathbb {R}$$
with the norm
$$|| \cdot ||_{\mathcal {M}}$$
introduced by Agram and Øksendal (Model uncertainty stochastic mean-field control.
arXiv:1611.01385v5
, [2]), and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (*time*-*advanced backward stochastic differential equations* (absdes), one of them with values in the space of bounded linear functionals on path segment spaces. As an application of our methods, we solve a memory mean–variance problem as well as a linear–quadratic problem of a memory process.

## Kalman Duality Principle for a Class of Ill-Posed Minimax Control Problems with Linear Differential-Algebraic Constraints

### Applied Mathematics & Optimization (2013-10-01) 68: 289-309 , October 01, 2013

In this paper we present Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and time-varying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable in the general case. We construct a minimizing sequence $\hat{u}_{\varepsilon}$ for the dual problem applying Tikhonov method. Finally we represent $\hat{u}_{\varepsilon}$ in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE.

## Contraction Options and Optimal Multiple-Stopping in Spectrally Negative Lévy Models

### Applied Mathematics & Optimization (2015-08-01) 72: 147-185 , August 01, 2015

This paper studies the optimal multiple-stopping problem arising in the context of the timing option to withdraw from a project in stages. The profits are driven by a general spectrally negative Lévy process. This allows the model to incorporate sudden declines of the project values, generalizing greatly the classical geometric Brownian motion model. We solve the one-stage case as well as the extension to the multiple-stage case. The optimal stopping times are of threshold-type and the value function admits an expression in terms of the scale function. A series of numerical experiments are conducted to verify the optimality and to evaluate the efficiency of the algorithm.