This work tackles a class of optimization problems in which fixing some well-chosen combinations of the variables makes the problem substantially easier to solve. We consider that the variables space may be partitioned into subsets that fix these combinations to given values, so that the restriction of the problem to any of the partition sets admits a tractable solution. Then, we exhibit a reformulation of the problem that consists in searching for the partition set index that minimizes the objective value of the solution to the restricted problem. We name partitioned optimization framework (POf) the formalization of this class of problems and this reformulation process. As we prove in this work, the POf allows solving the original problem by focusing on the reformulated problem: all solutions to the reformulated problem are partition indices for which the solution to the associated restricted problem is also a solution to the original problem. Second, we introduce a derivative-free partitioned optimization method (DFPOm) to efficiently solve problems that fit in the POf. We prove that the reformulated problem is nicely handled by derivative-free optimization (DFO) algorithms with a covering step. Then the DFPOm consists in solving the reformulated problem using such DFO algorithm with a covering step to obtain an optimal partition index, and to return the solution to the associated restricted problem as a solution to the initial problem. Finally, we illustrate how the POf allows solving some classes of problems. We first focus on an infinite-dimensional case, by solving analytically an optimal control problem that challenges standard methods from the literature. Then, we apply the DFPOm on a class of finite-dimensional problems called composite greybox problems, and we highlight the gain in numerical performance provided by the DFPOm by comparing it to two popular DFO solvers.

In this paper we address the minimum time problem for the double integrator, but here, in contrast with the classical version of this problem, the control is constrained to remain constant as long as the state belongs to a given region of the state space called loss-of-control region. This situation prevents switches from occurring in the loss-of-control region and, therefore, a new analysis has to be performed. For this purpose we prove an appropriate version of the Pontryagin maximum principle in which the necessary conditions comprise two key components. The first is an averaged Hamiltonian gradient condition to determine the optimal constant values of the control in the loss-of-control region. The second is, similarly to hybrid maximum principles found in the literature, that the costate admits discontinuity jumps at the interface between the loss-of-control region and its complement. We then highlight the theoretical use of these necessary conditions by solving analytically the minimum time problem for the double integrator with an illustrative loss-of-control region (precisely, the left vertical half-space). New behaviors are observed such as the lack of dynamic programming principle, of feedback expression and of saturation of the control constraint set. Finally we further illustrate these aspects by solving numerically the same minimum time problem for the double integrator but with two other illustrative loss-of-control regions (first a sloped half-space, then a disk).

Using advanced concepts and techniques from convex and variational analysis (such as the notion of twice epi-differentiability), we establish, under a set of appropriate conditions (including a polyhedricity assumption), that the solution to a parameterized nonlinear variational inequality associated with a positively homogeneous function is differentiable, and furthermore that its derivative is the solution to a corresponding complementarity problem. We illustrate our main result through a series of examples and counterexamples. In particular we introduce an example of a projection operator onto a nonempty closed convex cone that is not directionally differentiable, which is, to our best knowledge, new in the literature.

In this paper we consider a general control system involving a spatially heterogeneous dynamics. This means that the state space is partitioned into several disjoint regions and that each region has its own (smooth) control system. As a result, the dynamics discontinuously changes whenever the trajectory crosses an interface between two regions. In that spatially heterogeneous setting (and in contrast with the usual smooth case), a needle-like perturbation of the control may generate a perturbed trajectory that does not uniformly converge towards the nominal one, and may lead to the absence of a corresponding first-order variation vector. The first contribution of this paper is to illustrate this issue by means of a simple counterexample. Our second and main contribution is to provide a modified needle-like perturbation of the control (adapted to the spatially heterogeneous setting) which generates a perturbed trajectory that uniformly converges towards the nominal one, and leads to a corresponding first-order variation vector (which has the particularity of admitting a discontinuity jump at each interface crossing). This is made possible under several assumptions (including transverse crossing conditions), by introducing new tools such as auxiliary trajectories and auxiliary controls and by using a conic version of the implicit function theorem.

Consider, on the one part, a general nonlinear finite- dimensional optimal control problem and assume that it has a unique solution x*. On the other part, consider the sampled-data control version of it. Under appropriate assumptions, we prove that the optimal state of the sampled-data problem converges uniformly to x* as the norm of the partition tends to zero. Moreover, applying the Pontryagin maximum principle (PMP) to both problems, we prove that, if x* has a unique weak extremal lift with a costate p that is normal, then the costate of the sampled-data problem converges uniformly to p. In other words, under a normality assumption, control sampling commutes, at the limit of small partitions, with the application of the PMP.

The title of the present work is a nod to the paper ``The hybrid maximum principle is a consequence of Pontryagin maximum principle"" by Dmitruk and Kaganovich [ Systems Control Lett., 57 (2008), pp. 964--970]. We investigate a similar framework of hybrid optimal control problems that is also different from Dmitruk and Kaganovich's. Precisely, we consider a general control system that is described by a differential equation involving a spatially heterogeneous dynamics. In that context, the sequence of dynamics followed by the trajectory and the corresponding switching times are fully constrained by the state position. We prove with an explicit counterexample that the augmentation technique used by Dmitruk and Kaganovich cannot be fully applied to our setting, but we show that it can be adapted by introducing a new notion of local solution to classical optimal control problems and by establishing a corresponding Pontryagin maximum principle. Thanks to this method, we derive a hybrid maximum principle adapted to our setting, with a simple proof that does not require any technical tools (such as implicit function arguments) to handle the dynamical discontinuities.

This paper introduces a new step to the Direct Search Method (DSM) to strengthen its convergence analysis. By design, this so-called covering step may ensure that, for any refined point of the sequence of incumbent solutions generated by the resulting cDSM (covering dsm), the set of all evaluated trial points is dense in a neighborhood of that refined point. We prove that this additional property guarantees that all refined points are local solutions to the optimization problem. This new result holds true even for a discontinuous objective function, under a mild assumption that we discuss in details. We also provide a practical construction scheme for the covering step that works at low additional cost per iteration. Finally, we show that the covering step may be adapted to classes of algorithms differing from the DSM.

The aim of this work is to analyse a shape optimization problem in a mechanical friction context. Precisely we perform a shape sensitivity analysis of a Tresca friction problem, that is, a boundary value problem involving the usual linear elasticity equations together with the (nonsmooth) Tresca friction law on a part of the boundary. We prove that the solution to the Tresca friction problem admits a directional shape derivative which moreover coincides with the solution to a boundary value problem involving tangential Signorini's unilateral conditions. Then an explicit expression of the shape gradient of the Tresca energy functional is provided (which allows us to provide numerical simulations illustrating our theoretical results). Our methodology is not based on any regularization procedure, but rather on the twice epi-differentiability of the (nonsmooth) Tresca friction functional which is analyzed thanks to a change of variables which is well-suited in the two-dimensional case. The obstruction in the higher-dimensional case is discussed.

This paper addresses optimal control problems with loss control regions. In that context the state space is partitioned into disjoint subsets, referred to as regions, which are classified into two types: control regions and loss control regions. When the state belongs to a control region, the control is permanent (i.e. the control value is authorized to be modified at any time). On the contrary, when the state belongs to a loss control region, the control must remain constant as long as the state belongs to this region. The objective of this paper is twofold. First, we reformulate the above setting into a hybrid optimal control problem (with spatially heterogeneous dynamics) involving moreover a regionally switching parameter, and we prove a corresponding hybrid maximum principle: hence first-order necessary optimality conditions in a Pontryagin form are obtained. Second, this paper proposes a two-steps numerical scheme to solve optimal control problems with loss control regions. It is based on a direct numerical method (applied to a regularized problem) which initializes an indirect numerical method (applied to the original problem and based on the aforementioned necessary optimality conditions). This numerical approach is applied to several illustrative examples. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function f: ℝ→ℝ which satisfies all the assumptions required by the theorem but contradicts some of its conclusions. A corollary of this theorem is also affected by this counterexample. The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points (x_k)_k ∈ℕ converging to a point x_* where f is discontinuous, lower semicontinuous and whose objective function value f(x_*) is strictly less than lim _k→∞ f(x_k) . Moreover the dDSM generates trial points in only one of the continuity sets of f near x_* . This note also investigates the proof of the theorem to highlight the inexact statements in the original paper. Finally this work introduces a modification of the dDSM that allows, in usual cases, to recover the properties broken by the counterexample.

This paper investigates, without any regularization procedure, the sensitivity analysis of a mechanical friction problem involving the (nonsmooth) Tresca friction law in the linear elastic model. To this aim a recent methodology based on advanced tools from convex and variational analyses is used. Precisely we express the solution to the so-called Tresca friction problem thanks to the proximal operator associated with the corresponding Tresca friction functional. Then, using an extended version of twice epi-differentiability, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving tangential Signorini’s unilateral conditions. Finally our result is used to investigate and numerically solve an optimal control problem associated with the Tresca friction model.

This paper investigates, without any regularization or penalization procedure, a shape optimization problem involving a simplified friction phenomena modeled by a scalar Tresca friction law. Precisely, using tools from convex and variational analysis such as proximal operators and the notion of twice epi-differentiability, we prove that the solution to a scalar Tresca friction problem admits a directional derivative with respect to the shape which moreover coincides with the solution to a boundary value problem involving Signorini-type unilateral conditions. Then we explicitly characterize the shape gradient of the corresponding energy functional and we exhibit a descent direction. Finally numerical simulations are performed to solve the corresponding energy minimization problem under a volume constraint which shows the applicability.

In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called universal separating vector lemma is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method.

The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space H-1(omega) and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar, we show the differentiability of the solution to the parameterized Tresca-type problem and that its derivative satisfies a Signorini-type problem. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.

In this paper we consider a Mayer optimal control problem governed by a hybrid control system defined over a partition of the state space. We assume that the control system depends on a regionally switching parameter that remains constant in each region but that can change its value when the state position crosses interfaces. This new framework allows to deal, as a particular case, with control systems including non-control regions. In this paper our objective is to provide the corresponding necessary optimality conditions in a Pontryagin form. Our approach is based on a thorough sensitivity analysis of the hybrid control system under needle-like perturbations of the control and under convex perturbations of the parameter. To this aim we invoke implicit function arguments to deal with the interface crossings that are assumed to be transverse. The paper is concluded with a simple academic example showing that our framework allows to fill a gap in the literature.

This paper investigates the sensitivity analysis of a scalar mechanical contact problem described by a boundary value problem involving the Tresca’s friction law. The sensitivity analysis is performed with respect to right-hand source and boundary terms perturbations. In particular, the friction threshold involved in the Tresca’s friction law is perturbed, which constitutes the main novelty of the present work with respect to the existing literature. Hence, we introduce a parameterized Tresca friction problem and its solution is characterized by using the proximal operator associated with the corresponding perturbed nonsmooth convex Tresca friction functional. Then, by invoking the extended notion of twice epi-differentiability depending on a parameter, we prove the differentiability of the solution to the parameterized Tresca friction problem, characterizing its derivative as the solution to a boundary value problem involving Signorini unilateral conditions. Finally, numerical simulations are provided in order to illustrate our main result.

We consider a smooth control system that is subject to loss of control in the sense that the state space is partitioned into several disjoint regions and, in each region, either the system can be controlled, as usual, in a permanent way (that is, one can change the value of the control at any real time), or, on the contrary, the control has to remain constant from the entry time into the region until the exit time. The latter case corresponds to a non-control region. The objective of this paper is to state the necessary optimality conditions for a Mayer optimal control problem in such a setting of loss of control. Our main result is based on a hybrid maximum principle that takes into account a regionally switching parameter.

A piecewise constant Mayer cost function is used to model optimal control problems in which the state space is partitioned into several regions, each having its own Mayer cost value. In such a context, the standard numerical methods used in optimal control theory naturally fail, due to the discontinuities and the null gradients associated with the Mayer cost function. In this paper an hybrid numerical method, based on both derivative-free optimization and smooth optimization techniques, is proposed to solve this class of problems. Numerical simulations are performed on some standard control systems to show the efficiency of the hybrid method, where NOMAD and IPOPT are used as, respectively, derivative-free optimization and smooth optimization solvers.

Recent force-fatigue mathematical models in biomechanics [7] allow to predict the muscular force response to functional electrical stimulation (FES) and leads to the optimal control problem of maximizing the force. The stimulations are Dirac pulses and the control parameters are the pulses amplitudes and times of application, the number of pulses is physically limited and the model leads to a sampled data control problem. The aim of this article is to present and compare two methods. The first method is a direct optimization scheme where a further refined numerical discretization is applied on the dynamics. The second method is an indirect scheme: first-order Pontryagin type necessary conditions are derived and used to compute the optimal sampling times.

We revisit and extend the Riccati theory, unifying continuous-time linear-quadratic optimal permanent and sampled-data control problems, in finite and infinite time horizons. In a nutshell, we prove that:-- when the time horizon T tends to $+\infty$, one passes from the Sampled-Data Difference Riccati Equation (SD-DRE) to the Sampled-Data Algebraic Riccati Equation (SD-ARE), and from the Permanent Differential Riccati Equation (P-DRE) to the Permanent Algebraic Riccati Equation (P-ARE);-- when the maximal step of the time partition $\Delta$ tends to $0$, one passes from (SD-DRE) to (P-DRE), and from (SD-ARE) to (P-ARE).Our notations and analysis provide a unified framework in order to settle all corresponding results.