arXiv: Optimization and Control

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We study the ternary Ohta-Kawasaki free energy that has been used to model triblock copolymer systems. Its one-dimensional global minimizers are conjectured to have cyclic patterns. However, some physical experiments and computer simulations found triblock copolymers forming noncyclic lamellar patterns. In this work, by comparing the free energies of the cyclic pattern and some noncyclic candidates, we show that the conjecture does not hold for some choices of parameters. Our results suggest tha...

Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completion.

This paper is concerned with the low Tucker-rank tensor completion problem, which is about reconstructing a tensor \mathcal{T}\in\mathbb{R}^{n\times n\times n}of low multilinear rank from partially observed entries. We consider a manifold algorithm (i.e., Riemannian gradient method) for this problem and reveal an appealing implicit regularization phenomenon of non-convex optimization in low Tucker-rank tensor completion. More precisely, it has been rigorously proved that the iterates of the R...

Dynamic programming principle for classical and singular stochastic control with discretionary stopping

We prove the dynamic programming principle (DPP) in a class of problems where an agent controls a ddimensional diffusive dynamics via both classical and singular controls and, moreover, is able to terminate the optimisation at a time of her choosing, prior to a given maturity. The time-horizon of the problem is random and it is the smallest between a fixed terminal time and the first exit time of the state dynamics from a Borel set. We consider both the cases in which the total available fuel...

In this article, we consider transport networks with uncertain demands. Network dynamics are given by linear hyperbolic partial differential equations and suitable coupling conditions, while demands are incorporated as solutions to stochastic differential equations. For the demand satisfaction, we solve a constrained optimal control problem. Controls in terms of network inputs are then calculated explicitly for different assumptions. Numerical simulations are performed to underline the theoretic...

This paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin--Voigt damping $ u_{tt}(x,t)-[u_{x}(x,t)+b(x)u_{x,t}(x,t)]_{x}=0,\; x\in(-1,1),\; t>0, where (x)=0 on \in (-1,0], and (x)=x^\alpha>0 on \in (0,1) for alpha\in(0,1). It is known that the optimal decay rate of solution is ^{-2} in the limit case alpha=0, and exponential decay rate for alpha\ge 1. When alpha\in (0,1), the damping coefficient (x)$ is continuous, bu...

Physics-informed neural networks (PINNs) have recently become a popular method for solving forward and inverse problems governed by partial differential equations (PDEs). By incorporating the residual of the PDE into the loss function of a neural network-based surrogate model for the unknown state, PINNs can seamlessly blend measurement data with physical constraints. Here, we extend this framework to PDE-constrained optimal control problems, for which the governing PDE is fully known and the go...

Increasingly volatile electricity prices make simultaneous scheduling optimization for production processes and their energy supply systems desirable. Simultaneous scheduling needs to account for both process dynamics and binary on/off-decisions in the energy system and thus leads to challenging mixed-integer dynamic optimization problems. In this contribution, we propose an efficient scheduling formulation that consists of three parts: a linear scale-bridging model for the closed-loop process o...

In this paper we propose a resolvent splitting with minimal lifting for finding a zero of the sum of n\ge 2maximally monotone operators involving the composition with a linear bounded operator. The resolvent of each monotone operator, the linear operator, and its adjoint are computed exactly once in the proposed algorithm. In the case when the linear operator is the identity, we recover the resolvent splitting with minimal lifting developed in Malitsky-Tam (2021). We also derive a new resolve...

Park and Ride (P&R) facilities are car parks at which users can transfer to public transportation to reach their final destinations. Commuters can use P&R facilities or choose to travel by car to their destinations, and individual choice behavior is assumed to follow a logit model. The P&R facility location problem locates a fixed number of P&R facilities among potential locations, maximizing the number of users of P&R facilities. The problem is formalized in a nonlinear optimization problem usi...

This work aims to minimize a continuously differentiable convex function with Lipschitz continuous gradient under linear equality constraints. The proposed inertial algorithm results from the discretization of the second-order primal-dual dynamical system with asymptotically vanishing damping term considered by Bo\c t and Nguyen in [Bot, Nguyen, JDE, 2021], and it is formulated in terms of the Augmented Lagrangian associated with the minimization problem. The general setting we consider for the ...

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