DIFFERENTIAL OPTIMIZATION IN FINITE-DIMENSIONAL SPACES

. In this paper, a class of optimization problems coupled with differential equations in ﬁnite dimensional spaces are introduced and studied. An existence theorem of a Carath´eodory weak solution of the diﬀerential optimization problem is established. Furthermore, when both the mapping and the constraint set in the optimization problem are perturbed by two diﬀer-ent parameters, the stability analysis of the diﬀerential optimization problem is considered. Finally, an algorithm for solving the diﬀerential optimization problem is established.

1. Introduction. Let K be a subset of R m , g : K → R be a function. The following optimization problem: Minimize g(w) subject to w ∈ K has wide applications in engineering sciences, economics, finance, transportation and so on. There have been many publications devoted to optimization theory and applications [1,10,29,8,28,12,21,19,13,14]. Various kinds of methods have been developed for solving the optimization problem. For example, gradient flow technique [2] can be applied to find the optimal solution. In the approach an optimization problem is formulated as an ordinary differential equation (ODE) so that the solution of this ODE converges to an equilibrium point of the original problem. The approach has been improved and generalized by many authors in recent years. a unified gradient flow approach to nonlinear constrained optimization problems was presented in [24]. A novel hybrid descent method, consisting of a simulated annealing algorithm and a gradient-based method, was proposed to consider the optimal design of finite precision FIR filters in [26]. Gradient descent methods can be applied to solve integer programming problems in [6]. In addition, many important dynamical systems were modeled by optimization problem coupled with differential equations [5,18,23,22] and gradient flow technique can help us to study the dynamical optimization problem. It is known that the differentiability of objective function is an important assumption in the work. Based on the above researches, we would like to consider the dynamical optimization problem without the differentiability assumption in this paper. Consider the following differential optimization problem (DOP): Subject to w(t) ∈ K, : R n → R n×m and g : R n × R m → R are given mappings. In this paper, the solution set of the optimization problem (1) is denoted by SOL(K, g). (x(t), w(t)) defined on [0, T ] is called a Carathéodory weak solution of DOP (2) iff x(t) is an absolutely continuous function on [0, T ] and satisfies the differential equation for almost all t ∈ [0, T ] and w ∈ L 2 ([0, T ], R m ) and w(t) ∈ SOL(K, g(x(t), ·)) for every t ∈ [0, T ]. The Carathéodory weak solution set of the DOP (2) is denoted by SOL(DOP(2)).
Differential optimization problem such as the model (2) is seldom researched. Although it looks like an optimal control problem with x being the state and w being the control, there exist some differences between DOP (2) and the optimal control problems. The main difference lies in the following aspect: The control w in DOP (2) is the solution at time t for DOP (2), whose objective functions depend on the current state, and it is a pointwise optimization, while the control w in the optimal control problems is to minimize a performance function that is an integral function. There are many differential optimization problems arising in real worlds. For example, in static portfolio research, we use quadratic programming model to maximize the earning under the assumption that the risk is determined, and the quadratic programming is an optimization problem. In fact, the risk usually varies with time. We believe that the differential optimization problem is a more appropriate model characterizing the portfolio problem in a continuous-time system. Therefore, it is an interesting research to establish the existence result of a Carathéodory weak solution of DOP (2). Furthermore, there are many publications devoted to the stability analysis of static optimization problem. When the objective functions and the constraint sets were perturbed, researchers established various kinds of stability results of the parametric optimization problem (see for example [3,4,7,27]). The results can help us to observe the change of the optimal solution for parametric static optimization problem. However, there is few paper on the stability analysis of DOP (2). The main difference between DOP (2) and the static optimization problem lies in the following aspect: for DOP (2), we need to consider the state x and the optimal solution u at the same time, while for the static optimization problem, we only consider the optimal solution u. Therefore, it is more difficult to study DOP (2) compared with the static problem.
Let (Z 1 , d 1 ) and (Z 2 , d 2 ) be two metric spaces. Assume that a nonempty closed and convex set K ⊂ R n is perturbed by a parameter u, which varies over (Z 1 , d 1 ), that is, K : Z 1 ⇒ R n is a set-valued mapping with nonempty closed convex values. Let the objective function g : R n × R m → R be perturbed by a parameter v, which varies over (Z 2 , d 2 ), that is, g : R n × R m × Z 2 → R. We consider the parametric DOP: Subject to w(t) ∈ K(u), The Carathéodory weak solution set of the parametric DOP (3) is denoted by SOL(DOP(u, v)). The remainder of this paper is organized as follows. In section 2, we introduce some preliminary results. In section 3, we establish the existence result of a Carathéodory weak solution of DOP (2). In section 4, we study the stability analysis of DOP(3). In section 5, we give an algorithm for solving the differential optimization problem. In section 6, we give some numerical experiments to verify the validity of the proposed algorithm.

2.
Preliminaries. In this section, we will introduce some preliminary results.
Definition 2.1. Let Y , Z be topological spaces and G : Y ⇒ Z be a set-valued mapping with nonempty values. We say that G is (ii) lower semicontinuous at x 0 ∈ Y iff for any y 0 ∈ G(x 0 ) and any neighborhood N (y 0 ) of y 0 , there exists a neighborhood N (x 0 ) of x 0 such that We say that G is continuous at x 0 iff it is both upper and lower semicontinuous at x 0 . G is said to be continuous on Y iff it is both upper and lower semicontinuous at every point of Y .
: Ω ⇒ R n be an upper semicontinuous set-valued map with nonempty closed convex values. Suppose that there exists a scalar ρ F > 0 satisfying For every x 0 ∈ R n , the differential inclusionẋ ∈ F(t, x), x(0) = x 0 has a weak solution in the sense of Carathéodory.

Lemma 2.3. [17]
Let h : Ω × R m → R n be a continuous function and U : Ω ⇒ R m be a closed set-valued map such that for some constant η U > 0, Let v : [0, T ] → R n be a measurable function and x : 3. Existence of the solution for DOP. In Theorem 3.1, we establish the existence conclusion of a Carathéodory weak solution.
Theorem 3.1. Let f : R n → R n , B : R n → R n×m be two Lipschitz continuous mappings, g : R n × R m → R be a convex function, K be a compact and convex sets in R m , Then for every x 0 ∈ R n , the following differential inclusion: has a weak solution in the sense of Carathéodory. Furthermore, DOP(2) has a weak solution in the sense of Carathéodory.
Proof. The convexity of g on R n × R m implies that g is continuous on R n × R m . By the conditions that g is continuous on R n × R m and K is a compact subset of R m , it implies that for any q ∈ R n , the following optimization problem: has a solution. The continuity of g implies that for every q ∈ R n , SOL(K, g(q, ·)) is closed. Now we prove that SOL(K, g(q, ·)) is convex. Let g min denote the minimum of the objective function g(q, ·) on K. For every w 1 , w 2 ∈ SOL(K, g(q, ·)) and λ ∈ [0, 1], the convexity of g implies that It follows that g(q, λw 1 +(1−λ)w 2 ) = g min and so λw 1 +(1−λ)w 2 ∈ SOL(K, g(q, ·)). Next we prove that F is upper semicontinuous, it suffices to prove that F is closed from the compactness of K. Take a sequence {(t k , x k )} ∈ [0, T ] × R n . Let w k ∈ SOL(K, g(q, ·)) for every k = 1, 2 · · · . Assume that The compactness of K implies that there exists a subsequence of {w k }, which is denoted again by {w k }, such that w k → w 0 . The continuity of g implies that w 0 ∈ SOL(K, g(x 0 , ·)). From the uniqueness of limitation and the continuity of f and B, it follows that Therefore, F is closed and so upper semicontinuous.
Then we prove that the linear growth property of F. Since f and B are Lipschitz continuous, it implies that there exist ρ f > 0 and ρ B > 0 such that So F satisfies the linear growth property.
Based on the above discussion, it implies that F is upper semicontinuous with nonempty closed and convex values, and F satisfies the linear growth property. From Lemma 2.2, it follows that differential inclusion (5) To prove the closeness of SOL(DOP(u, v)), it suffices to prove that (x 0 , w 0 ) ∈ SOL(DOP(u 0 , v 0 )). By the assumption that (x n , w n ) ∈ SOL(DOP(u n , v n )), we have (i) dx n (t) dt = f (x n (t)) + B(x n (t))w n (t), (ii) for every t ∈ [0, T ] andw n ∈ K(u n ), (iii) the initial condition x n (0) = x 0 . (8) The absolute continuity of x n (t) implies that the equation (6) is equivalent to the following relation: for any 0 ≤ s ≤ t ≤ T , As (x n , w n ) → (x 0 , w 0 ), it follows from the continuity of f and B that for any 0 ≤ s ≤ t ≤ T , Next we prove that w 0 (t) ∈ SOL(K(u 0 ), g(x 0 (t), ·, v 0 )) for every t ∈ [0, T ]. Since w n → w 0 , it follows that for every t ∈ [0, T ], w n (t) → w 0 (t). The upper semicontinuity of K and w n (t) ∈ K(u n ) imply that w 0 (t) ∈ K(u 0 ) for every t ∈ [0, T ]. Assume for the sake of contradiction that there exists t ∈ [0, T ] such that w 0 (t) / ∈ SOL(K(u 0 ), g(x 0 (t), ·, v 0 )), Then there existsw 0 ∈ K(u 0 ) such that By the lower semicontinuity of K, it implies that there existw n ∈ K(u n ) such that w n →w 0 and so (x n ,w n , v n ) → (x 0 ,w 0 , v 0 ). The continuity of g on R n × R m × Z 2 implies that g(x n (t),w n , v n ) → g(x 0 (t),w 0 , v 0 ). Then there exists N 1 > 0 with n > N 1 such that Since (x n , w n , v n ) → (x 0 ,w 0 , v 0 ), it implies that Then there exists N 2 > 0 with n > N 2 such that It follows from (10), (11) and (12) that for any n > max{N 1 , N 2 }, and so w n (t) / ∈ SOL(DOP(u n , v n )), this is a contradiction to (7). Then for every t ∈ [0, T ] andw 0 ∈ K(u 0 ), Since x n → x 0 , it follows that x 0 (0) = x 0 . (14) Therefore, (9), (13)and (14) imply that (x 0 , w 0 ) ∈ SOL(DOP(u 0 , v 0 )). As (u 0 , v 0 ) is taken arbitrarily on Z 1 × Z 2 , it deduces that SOL(DOP(u, v)) is closed on Z 1 × Z 2 . This completes the proof.
5. An algorithm for DOP. Based on the numerical method to derive the solution for differential variational inequality [17], we consider the algorithm for DOP (2). It begins with the division of the time interval [0, T ] into N l + 1 subintervals: 0 = t l,0 < t l,1 < · · · < t l,N l < t l,N l +1 = T, where l > 0 and (N l + 1)l = T and t l,i+1 = t l,i + l, for all i = 0, 1, · · · , N l .
Theorem 5.1. Let f : R n → R n , B : R n → R n×m be two Lipschitz continuous mappings, let f (R n ) and B(R n ) be bounded, and let K be a bounded, closed and convex subset of R n . Then there is a sequence {l v } ↓ 0 such that the following two limits exist: x lv → x and w lv w in L 2 ([0, T ], K), where x lv and w lv are defined by (17), and denotes the weak convergence. Furthermore, assume that the following conditions hold: (i) for any w ∈ K, g(·, w) is continuous on R n , (ii) for any x ∈ R n , g(x, ·) is strongly continuous on L 2 ([0, T ], K), then all such limits (x, w) are weak solutions of DOP(2).
Proof. Since x l,i+1 = x l,i + l(f (x l,i ) + B(x l,i )w l,i ) and the boundedness of f , B and K, it follows that there exists M > 0 such that This implies that {x l (t)} is equicontinuous and uniform bounded. By Arzela-Ascoli theorem, there exists a sequence {l v } ↓ 0 such that {x lv } converges to a function x with respect to the norm x 1 = sup x(t) . Following a similar way to the proof of Theorem 7.1 in [17], we have where O(t) denotes a function satisfying lim t↓0 O(t) t < ∞. Then for any 0 ≤ s ≤ t ≤ T , we have As l v → 0, it follows that, The boundedness of K implies that {w lv } is uniform bounded. By Alaoglu's theorem, it follows that the sequence {w lv } has a weak * limit w. The reflexive Banach space L 2 ([0, T ], K) implies that weak * convergent sequences are also weakly convergent sequences. In addition, we have and for every t ∈ [0, T ], g(x lv (t),ŵ) − g(x lv (t), w lv (t)) ≥ 0, ∀ŵ ∈ K.
Therefore, it follows from (18) and (19) that (x, w) is a weak solutions of the DOP(2).
6. Numerical experiments. In this section, we provide some examples to verify the validity of algorithm, which has been introduced in Section 5. We firstly study a differential optimization problem without a perturbed parameter in Example 6.1.
Step 0. It begins with the division of the time interval [0, 3] into 30 subintervals: with each of length l = 0.1.
Step 1. Let x l,0 = 0.1. Compute w l,0 which satisfies the following optimization problem, Step 2. Let x l,i+1 = x l,i + l(|x l,i − 0.3| + 3x l,i w l,i + 5), and let w l,i+1 be the solution of the following optimization problem, By the recursion, for i = 0, 1, 2, . . . , 30, the numerical results are shown in Figure 1. In Example 6.2, we consider the differential optimization problem when the objective function is perturbed by a parameter v.
Following a similar algorithm to Example 6.1, the numerical results of Example 6.2 are shown in Figure 2. Observe the trajectories of x(t) and w(t) when the parameters v 1 = 1, v 2 = 0.9, v 3 = 0.7, v 4 = 0.4, we can find that (   Furthermore, in Example 6.3, we consider the differential optimization problem when the constraint set is perturbed by a parameter u.
Following a similar algorithm to Example 6.1, the numerical results of Example 6.3 are shown in Figure 3.
Observe the trajectories of x(t) and w(t) when the parameters u 1 = 1, u 2 = 0.33, u 3 = 0.3, u 4 = 0.1, we can find that (x u , w u ) → (x 1 , w 1 ) as u → 1, where (x u , w u ) denotes a Carathéodory weak solution of DOP (23). The concluions of Example 6.2 and 6.3 are consistent with the conclusion of Theorem 4.1 that the Carathéodory weak solution set mapping is closed. 7. Conclusions. In this paper, a class of differential optimization problems have been introduced and studied. The main contributions in this paper include establishing the existence theorem of a Carathéodory weak solution of the differential optimization problem, studying the stability analysis of differential optimization problem, and establishing an algorithm for solving the differential optimization problem. As future extensions, the research on how realizable this approach is for dynamical portfolio problem and other real world large scale problems should be given. In addition, differential vector optimization problem, consisting of a system  of differential equation and vector optimization problem, can also be studied, since the model can be used to study the fermentation dynamics problem [20], human migration networks [16] and so on.