Optimization and Control (math.OC)

Abstract: In this research, we present an analysis of the optimal investment, consumption, and life insurance acquisition problem for a wage earner with partial information. Our study considers the non-linear filter case where risky asset prices are correlated to the factor processes under constant relative risk aversion (CRRA) preferences. We introduce a more general framework with an incomplete market, random parameters adapted to the Brownian motion filtration, and a general factor process with a non-linear state estimation and a correlation between the state process (risky asset prices) and the factor process. To address the wage earner's problem, we formulate it as a stochastic control problem with partial information where the risky assets prices are correlated to the factor processes. Our framework is extensive since the non-linear filter applied to the linear case gives a more robust result than the Kalman filter. We obtain the non-linear filter through the Zakai equation and derive a system of the Hamilton-Jacobi-Bellman (HJB) equation and two backward stochastic differential equations (BSDE). We establish the existence and uniqueness of the solution, prove the verification theorem, and construct the optimal strategy.

Abstract: We consider a nonlinear discrete stochastic control system, and our goal is to design a feedback control policy in order to lead the system to a prespecified state. We adopt a stochastic approximation viewpoint of this problem. It is known that by solving the corresponding continuous-time deterministic system, and using the resulting feedback control policy, one ensures almost sure convergence to the prespecified state in the discrete system. In this paper, we adopt such a control mechanism and provide its finite-sample convergence bounds whenever a Lyapunov function is known for the continuous system. In particular, we consider four cases based on whether the Lyapunov function for the continuous system gives exponential or sub-exponential rates and based on whether it is smooth or not. We provide the finite-time bounds in all cases. Our proof relies on constructing a Lyapunov function for the discrete system based on the given Lyapunov function for the continuous system. We do this by appropriately smoothing the given function using the Moreau envelope. We present numerical experiments corresponding to the various cases, which validate the rates we establish.

Abstract: In this article, we derive necessary and sufficient conditions for the existence of solutions to state-constrained continuity inclusions in Wasserstein spaces whose right-hand sides may be discontinuous in time. These latter are based on fine investigations of the infinitesimal behaviour of the underlying reachable sets, through which we show that up to a negligible set of times, every admissible velocity of the inclusion can be approximately realised as the metric derivative of a solution of the dynamics, and vice versa. Building on these results, we are able to establish necessary and sufficient geometric conditions for the viability and invariance of stationary and time-dependent constraints, which involve a suitable notion of contingent cones in Wasserstein spaces, presented in ascending order of generality. We then close the article by exhibiting two prototypical examples of constraints sets appearing in applications for which one can compute relevant subfamilies of contingent directions.

Abstract: In this article we consider the sparse solutions of the tensor complementarity problem (TCP) which are the solutions of the smallest cardinality. We establish a connection between the least element of the feasible solution set of TCP and sparse solution for $Z$-tensor. We propose a $p$ norm regularized minimization model when $p\in (0,1)$ and show that it can approximate sparse solution applying the regularization of parameter. Keywords: Tensor complementarity problem, sparse solution, $l_p$ regularized minimization, $Z$-tensor.

Abstract: In this paper, a descent method for nonsmooth multiobjective optimization problems on complete Riemannian manifolds is proposed. The objective functions are only assumed to be locally Lipschitz continuous instead of convexity used in existing methods. A necessary condition for Pareto optimality in Euclidean space is generalized to the Riemannian setting. At every iteration, an acceptable descent direction is obtained by constructing a convex hull of some Riemannian $\varepsilon$-subgradients. And then a Riemannian Armijo-type line search is executed to produce the next iterate. The convergence result is established in the sense that a point satisfying the necessary condition for Pareto optimality can be generated by the algorithm in a finite number of iterations. Finally, some preliminary numerical results are reported, which show that the proposed method is efficient.

Abstract: We study the problem of optimally routing plug-in electric and conventional fuel vehicles on a city level. In our model, commuters selfishly aim to minimize a local cost that combines travel time, from a fixed origin to a desired destination, and the monetary cost of using city facilities, parking or service stations. The traffic authority can influence the commuters' preferred routing choice by means of personalized discounts on parking tickets and on the energy price at service stations. We formalize the problem of designing these monetary incentives optimally as a large-scale bilevel game, where constraints arise at both levels due to the finite capacities of city facilities and incentives budget. Then, we develop an efficient decentralized solution scheme with convergence guarantees based on BIG Hype, a recently-proposed hypergradient-based algorithm for hierarchical games. Finally, we validate our model via numerical simulations over the Anaheim's network, and show that the proposed approach produces sensible results in terms of traffic decongestion and it is able to solve in minutes problems with more than 48000 variables and 110000 constraints.

Abstract: In automated driving, risk describes potential harm to passengers of an autonomous vehicle (AV) and other road users. Recent studies suggest that human-like driving behavior emerges from embedding risk in AV motion planning algorithms. Additionally, providing evidence that risk is minimized during the AV operation is essential to vehicle safety certification. However, there has yet to be a consensus on how to define and operationalize risk in motion planning or how to bound or minimize it during operation. In this paper, we define a stochastic risk measure and introduce it as a constraint into both robust and stochastic nonlinear model predictive path-following controllers (RMPC and SMPC respectively). We compare the vehicle's behavior arising from employing SMPC and RMPC with respect to safety and path-following performance. Further, the implementation of an automated driving example is provided, showcasing the effects of different risk tolerances and uncertainty growths in predictions of other road users for both cases. We find that the RMPC is significantly more conservative than the SMPC, while also displaying greater following errors towards references. Further, the RMPCs behavior cannot be considered as human-like. Moreover, unlike SMPC, the RMPC cannot account for different risk tolerances. The RMPC generates undesired driving behavior for even moderate uncertainties, which are handled better by the SMPC.

Abstract: This paper studies the problem of coordinating a group of $n$th-order integrator systems. As for the well-studied conventional consensus problem, we consider linear and distributed control with only local and relative measurements. We propose a closed-loop dynamic that we call serial consensus and prove it achieves $n$th order consensus regardless of model order and underlying network graph. This alleviates an important scalability limitation in conventional consensus dynamics of order $n\ge 2$, whereby they may lose stability if the underlying network grows. The distributed control law which achieves the desired closed loop dynamics is shown to be localized and obey the limitation to relative state measurements. Furthermore, through use of the small-gain theorem, the serial consensus system is shown to be robust to both model and feedback uncertainties. We illustrate the theoretical results through examples.

Abstract: In this contribution, we study an optimal control problem for the celebrated nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins potential in the three-dimensional setting. The control enters the governing state system in a nonlinear fashion in the form of a prescribed solenoidal, that is a divergence-free, vector field, whereas the cost functional to be minimized is of tracking-type. The novelties of the present paper are twofold: in addition to the control application, the intrinsic difficulties of the optimization problem forced us to first establish new regularity results on the nonlocal Cahn-Hilliard equation that were unknown even without the coupling with a velocity field and are therefore of independent interest. This happens to be shown using the recently proved separation property along with ad hoc H\"older regularities and a bootstrap method. For the control problem, the existence of an optimal strategy as well as first-order necessary conditions are then established.

Abstract: We study model predictive control (MPC) problems for stochastic LTI systems, where the noise distribution is unknown, compactly supported, and only observable through a limited number of i.i.d. noise samples. Building upon recent results in the literature, which show that distributional uncertainty can be efficiently captured within a Wasserstein ambiguity set, and that such ambiguity sets propagate exactly through the system dynamics, we start by formulating a novel Wasserstein Tube MPC (WT-MPC) problem, with distributionally robust CVaR constraints. We then show that the WT-MPC problem: (1) is a direct generalization of the (deterministic) Robust Tube MPC (RT-MPC) to the stochastic setting; (2) through a scalar parameter, it interpolates between the data-driven formulation based on sample average approximation and the RT-MPC formulation, allowing us to optimally trade between safety and performance; (3) admits a tractable convex reformulation; and (4) is recursively feasible. We conclude the paper with a numerical comparison of WT-MPC and RT-MPC.

Abstract: In robust optimisation problems the objective function consists of an average over (an ensemble of) uncertain parameters. Ensemble optimisation (EnOpt) implements steepest descent by estimating the gradient using linear regression on Monte-Carlo simulations of (an ensemble of) control parameters. Applying EnOpt for robust optimisation is costly unless the evaluations over the two ensembles are combined, i.e. 'paired'. Here, we provide a new and more rigorous perspective on the stochastic simplex approximate gradient (StoSAG) used in EnOpt, explaining how it addresses detrimental cross-correlations arising from pairing by only capturing the variability due to the control vector, and not the vector of uncertain parameters. A few minor variants are derived from a generalised derivation, as well as a new approach using decorrelation. These variants are tested on linear and non-linear toy gradient estimation problems, where they achieve highly similar accuracy, but require a very large ensemble size to outperform the non-robust approach when accounting for variance and not just bias. Other original contributions include a discussion of the particular robust control objectives for which EnOpt is suited, illustrations, a variance reduction perspective, and a discussion on the centring in covariance and gradient estimation.

Abstract: Nonnegativity certificates can be used to obtain tight dual bounds for polynomial optimization problems. Hierarchies of certificate-based relaxations ensure convergence to the global optimum, but higher levels of such hierarchies can become very computationally expensive, and the well-known sums of squares hierarchies scale poorly with the degree of the polynomials. This has motivated research into alternative certificates and approaches to global optimization. We consider sums of nonnegative circuit polynomials (SONC) certificates, which are well-suited for sparse problems since the computational cost depends on the number of terms in the polynomials and does not depend on the degrees of the polynomials. We propose a method that guarantees that given finite variable domains, a SONC relaxation will yield a finite dual bound. This method opens up a new approach to utilizing variable bounds in SONC-based methods, which is particularly crucial for integrating SONC relaxations into branch-and-bound algorithms. We report on computational experiments with incorporating SONC relaxations into the spatial branch-and-bound algorithm of the mixed-integer nonlinear programming framework SCIP. Applying our strengthening method increases the number of instances where the SONC relaxation of the root node yielded a finite dual bound from 9 to 330 out of 349 instances in the test set.

Abstract: This paper is concerned with stability of the ball for a class of isoperimetric problems under convexity constraint. Considering the problem of minimizing $P+\varepsilon R$ among convex subsets of $\mathbb{R}^N$ of fixed volume, where $P$ is the perimeter functional, $R$ is a perturbative term and $\varepsilon>0$ is a small parameter, stability of the ball for this perturbed isoperimetric problem means that the ball is the unique (local, up to translation) minimizer for any $\varepsilon$ sufficiently small. We investigate independently two specific cases where $\Omega\mapsto R(\Omega)$ is an energy arising from PDE theory, namely the capacity and the first Dirichlet eigenvalue of a domain $\Omega\subset\mathbb{R}^N$. While in both cases stability fails among all shapes, in the first case we prove (non-sharp) stability of the ball among convex shapes, by building an appropriate competitor for the capacity of a perturbation of the ball. In the second case we prove sharp stability of the ball among convex shapes by providing the optimal range of $\varepsilon$ such that stability holds, relying on the \emph{selection principle} technique and a regularity theory under convexity constraint.

Abstract: The Bass diffusion equation is a well-known and established modeling approach for describing new product adoption in a competitive market. This model also describes diffusion phenomena in various contexts: infectious disease spread modeling and estimation, rumor spread on social networks, prediction of renewable energy technology markets, among others. Most of these models, however, consider a deterministic trajectory of the associated state variable (e.g., market-share). In reality, the diffusion process is subject to noise, and a stochastic component must be added to the state dynamics. The stochastic Bass model has also been studied in many areas, such as energy markets and marketing. Exploring the stochastic version of the Bass diffusion model, we propose in this work an approximation of (stochastic) optimal control surfaces for a continuous-time problem arising from a $2\times2$ skew symmetric evolutionary game, providing the stochastic counter-part of the Fourier-based optimal control approximation already existent in the literature.

Abstract: For time inconsistent optimal control problems, a quite popular approach is the equilibrium approach, taken by the sophisticated agents. In this short note we construct a deterministic continuous time example where the unique equilibrium is dominated by another control. Therefore, in this situation it may not be wise to take the equilibrium strategy.