Reply to Comments on Our Paper "On the Relation Between Two Approaches to Necessary Optimality Conditions in Problems with State Constraints"

TL;DR

This paper addresses critical claims about a previous work on optimality conditions with state constraints, providing clarifications and rebuttals to advance understanding in the field.

Contribution

It offers a detailed response to critiques of prior research, clarifying the relationship between two approaches to necessary optimality conditions.

Findings

Clarifies misunderstandings about the original paper

Reaffirms the validity of the proposed approaches

Provides insights into the relation between different optimality conditions

Abstract

We consider some critical claims concerning our above paper, and reply to these claims.

Full text

      Here we reply to these claims.

Reply to Comments by Karamzin

First of all, a highly biased style of these comments leaps to the eye. However, let us set it aside and concentrate on mathematics. One can see that the author’s claims can be essentially reduced to the following: a) lack of novelty (that our results ”can simply be derived from the already known results in the literature”), and b) incompleteness of the result (stationarity conditions instead of Maximum Principle). Let us make sense of this.

As was clearly said in our paper [2, Remark 9.1], its novelty is a complete realization of the idea of obtaining optimality conditions (in this case, conditions of stationarity) in problems with a state constraint by differentiating the state constraint on the contact interval and thus by passing to a problem with a mixed constraint of the equality type. This idea was first proposed by Gamkrelidze [3], but in our paper it was realized, for a specific class of problems, by another approach. It needs to be emphasized that, rather than relying on Gamkrelidze’s result, in order to obtain optimality conditions in the form of Dubovitskii and Milyutin [4] we realized his idea  for obtaining these conditions. (Perhaps, this should have been said in [2] more clearly.) Our method is based on the replication of the state and control variables in accordance with the number of qualitatively different subarcs of the reference trajectory (interior or boundary w.r.t. the state constraint). This replication trick is quite natural and has been known for a long time (we give several references in [2]), but to our knowledge it has never been used in the given situation with complete details and clarity (and with a new feature consisting of a ”two-stage varying” of the reference trajectory). Despite the author’s assertion, it is not the ”reduction to $v-$problem”  in the sense of the $v-$problem proposed by Dubovitskii and Milyutin [4] and used in [5, Sec. 2.5].  The book [5] does not contain the replication method.

Note that this replication trick turned out to be effective in some other problems as well, such as problems with the control system of hybrid type, with intermediate constraints, and with a variable structure, where it allows to reduce these problems to a standard optimal control problem and to apply the already known optimality conditions, but rather surprisingly, it was missed even by highly prominent mathematicians, who obtained optimality conditions in these problems by performing all the heavy procedure of variational analysis (see comments and references in [6, 7]). Therefore, in our opinion, this method is quite worthy of attention.

Now, let us dwell on the paper by Neustadt [8]. The author of the note claims that its results allow one to make a passage from the conditions of Gamkrelidze (GamC) to those of Dubovitskii and Milyutin (DMC) by a change of the costate variable, proposed in [8]. However, this is not true, because the proposed passage contains a vicious circle. The matter is that Neustadt’s paper is not based either on the result of Gamkrelidze, or on his idea of differentiating the state constraint on the contact interval. His approach is in fact close to that of Dubovitskii and Milyutin;  namely, just like they, he considers the state constraint as an inequality constraint in the space of continuous function, and exactly from here, making the linearization of all the constraints and the cost, and applying then the Farkas-Minkowski theorem, he directly obtains a sign-definite measure possessing all the required properties. He clearly says (p. 134) that his conditions coincide  with DMC and imply  GamC. In fact, he makes a passage from DMC to GamC. (The same passage is later performed by Arutyunov, Karamzin, and Pereira in [9], with no reference to [8].) But the reverse passage GamC $\to$ DMC (by the reverse transformation of the costate) is valid only if the measure in GamC is completely sign-definite (and in this case one does not need to rely on paper [8], since the transformation, as well as the calculations in [1], is quite simple), whereas GamC do not provide the information about the sign of the measure at its atoms.

Thus, in the claim concerning novelty Mr. Karamzin misleads the reader by incorrect interpretations of methods and results in [5, 8, 9].

Now let’s address the alleged incompleteness of our result. Note again that we confined the study to the stationarity conditions (i.e. necessary first order conditions) for the extended weak minimality. Such conditions constitute an important stage in the study of any optimization problem. (Recall here the Fermat condition $f^{\prime}(x_{0})=0$ in the problem $f(x)\to\min,$ the Euler and Euler-Lagrange equations in the calculus of variations, the Lagrange multipliers rule in the nonlinear programming, etc.)  Like any classical necessary first order conditions, they are not complete in the sense that they can be strengthened (e.g. by second order conditions). But as the stationarity conditions  they are complete, since they are equivalent, loosely speaking, to the non-negativity of the cost derivative in all admissible directions, and because of this their importance remains undisturbed even if they are strengthened by other conditions. (The Weierstrass necessary condition does not invalidate the Euler equation!)  The issue of obtaining Maximum Principle by the given approach  was not considered in our paper, it requires additional study.   So, this claim of the author is also inadequate.

Obviously, the presence of extended weak minimality implies the strong minimality in an $\varepsilon$-tube around the reference control, and hence, the fulfillment of Maximum Principle in this $\varepsilon$-tube, which in turn implies the absence of atoms of the measure at the junction points with the state boundary. In view of this, our sentence in [2, p. 407], given below in bold italics, is, indeed, not correctly formulated:

”Studies show … that in case of strong (or at least Pontryagin type [16,17]) minimality, the adjoint variable and measure do not have jumps under condition (2). However, this result is not, in general, valid in the case of extended weak minimality (the reason is that one cannot rely upon the maximality of Pontryagin function w.r.t. u, having in disposal only the stationarity of the extended Pontryagin function).”

Here we should have said more precisely:  ”However, this result is not, in general, valid in the case of stationarity”. In fact, this is then said in brackets. Note also that in p. 410 we give a correct resume:

”Thus, the stationarity conditions do not guarantee the absence of atoms, while, according to [5,15,19–23], the maximum principle does.”  We hope that the attentive reader will be able to understand this point properly.

Having said that ”the technique of reducing to a mixed constrained problem is obviously too restrictive as important information on the admissible trajectories subject to state constraints is lost by this transition”, Mr. Karamzin missed the fact that this is just the first stage of our variation procedure. At the second stage we use variations that go inside the state constraints.

As concerns an omission in our assumptions, of course the data functions $f^{\prime}_{u}$ and $g^{\prime}_{u}$ should be assumed Lipschitz continuous, not just continuous.

The author’s claims concerning ”wrong” citations and ”confusing” title present just his personal opinion.