Direct optimization methods for solving a complex state constrained optimal control problem in microeconomics

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1 Direct optimization methods for solving a complex state constrained optimal control problem in microeconomics Helmut Maurer a Hans Josef Pesch b a Universität Münster, Institut für Numerische und Angewandte Mathematik Einsteinstr. 62, Münster, Germany, maurer@math.uni-muenster.de b Universität Bayreuth, Fakultät für Mathematik und Physik, Universitätstr. 3, 9544 Bayreuth, Germany, hans-josef.pesch@uni-bayreuth.de Abstract We analyze and solve a complex optimal control problem in microeconomics which has been investigated earlier in the literature. The complexity of the control problem originates from four control variables appearing linearly in the dynamics and several state inequality constraints. Thus the control problem offers a considerable challenge to the numerical analyst. We implement a hybrid optimization approach which combines two direct optimization methods. The first step consists in solving the discretized control problem by nonlinear programming methods. The second step is a refinement step where, in addition to the discretized control and state variables, the junction times between bang bang, singular and boundary subarcs are optimized. The computed solutions are shown to satisfy precisely the necessary optimality conditions of the Maximum Principle where the state constraints are directly adjoined to the Hamiltonian. Despite the complexity of the control structure, we are able to verify sufficient optimality conditions which are based on the concavity of the maximized Hamiltonian. Key words: microeconomic control model, control of stock, labor and capital, state inequality constraints, direct optimization methods, bang bang and singular control, verification of necessary and sufficient conditions PACS: 49J15, 49K15, 58E17, 65K5 1 Introduction The well known microeconomic concern model of Lesourne, Leban [1] involves only capital flows as control and state variables. Koslik, Breitner [8] and Winderl, Naumer [16] have developed an extended concern model that Preprint submitted to Elsevier 13 September 27

2 includes the production and employment sector. Besides its economic interest, the optimal control problem constitutes a considerable numerical challenge, since it comprises four control variables appearing linearly in the dynamics and several pure state inequality constraints. In [8,16], a hybrid numerical approach has been developed to determine the complicated control switching structure. First, a discretized version of the control problem is solved by nonlinear programming methods. This method yields reliable estimates for the control and state variables on a fixed grid. The second step is a refinement step, where the control and state estimates are used in the so called indirect method which requires the solution of a boundary value problem (BVP) for the state and adjoint variables. For the concern model, it is extremely difficult to set up the BVP due to the presence of pure state constraints. For this reason, authors [8,16] have substituted the active state constraints by suitable mixed control state constraints that are better tractable in the BVP formulation. The purpose of the paper is twofold. First, we discuss direct optimization methods that provide solutions which satisfy precisely the Maximum Principle for state constrained optimal control problems. The second goal is to show that the computed solution satisfies a suitable type of sufficient optimality conditions. The organisation of the paper is as follows. The concern model is presented in Section 2. In Section 3, necessary optimality conditions are discussed which are based on a Maximum Principle, where the state constraints are directly adjoined to the Hamiltonian. In Section 4, we present a hybrid optimization approach to solve the state constrained control problem. The first step is similar to that in [8,16] and differs only in that we apply the large scale optimization methods developed by Büskens [2,3] and Wächter [15]. The second step is different from the one in [8,16]. Instead of trying to solve the BVP of the Maximum Principle, we optimize simultaneously the switching and junction times between bang bang, singular and boundary arcs and the discretized control variables; cf. [5,13]. The computed control and state variables satisfy the Maximum Principle with high accuracy. Finally, in Section 5 we show that the computed solutions satisfy a suitable type of sufficient conditions. 2 Optimal control model for a concern with four control variables appearing linearly and state constraints The microeconomic control model discussed in Koslik, Breitner [8] and Winderl, Naumer [16] has six state variables and four control variables x = (S, L, Y, X, X m, X r ) IR 6, u = (S c, L c, Y c, I) IR 4, 2

3 which have the following meaning. The stock S(t) is controlled by S c (t); the number L(t) of employees is controlled by the employment rate L c (t); the capital consists of loan capital Y (t) and equity capital X(t); the control Y c (t) describes the borrowing of loan capital while the owners of the equity capital choose by means of the investment control I(t) between an investment within the concern and an alternative investment X m (t); the risk premium X r (t) serves as a reserve fund for the safety of the capital owners; the risk premium is denoted by ρ r (t). All parameters and functions appearing in the following quantities and differential equations are summarized in Table 1. The production function (output) is assumed to be of Cobb Douglas type: F (x) = F (L, Y, X) = α(x + Y ) α K L α L. (1) Then the profit (gain) of the concern is given by G(x, u, t) = 1 d(t) [ p(t)(f (x) S c) σs ω(t)l ] ρ K (t)y δ(x + Y ). (2) The discount rate d(t) is defined as the solution of the differential equation d(t) = d(t)ln(1 + i(t)), (3) where i(t) is the periodic inflation rate specified in Table 1. Note that in contrast to the presentation in [8,16], we do not treat d(t) as a state variable. In section 5, this viewpoint will allow us to apply sufficient optimality conditions. The dynamics is governed by differential equations with fixed initial values, ẋ(t) = f(x(t), u(t), t), x() = x, (4) which are given explicitly by Ṡ(t) = S c (t), S() = 1, L(t) = L c (t), L() = 3, Ẏ (t) = Y c (t), Y () = 5, Ẋ(t) = I(t) + (1 τ) [ G(x(t), u(t), t) ρ r (t)x(t) ], X() = 1, Ẋ m (t) = I(t) + (1 τ)ρ m (t)x m (t), X m () = 45, Ẋ r (t) = (1 τ)ρ r (t)x(t), X r () =. (5) 3

4 Notation Formula / Value Meaning t f 1 time horizon in years F (x) α(x + Y ) α K L α L production function (output) α 1 parameter in production function α K.35 elasticity of total capital K = X + Y α L.5 elasticity of labor k l 8 duration of economic cycle k p (t) π 2 + 2π k l t position in economic cycle ρ K (t) sin k p (t) loan interest rate ρ m (t) sin k p (t) current yield ρ r (t) ρ K (t).5 risk premium rate ρ low r (t) 2 3 (ρ K(t).8) +.2 risk premium rate for daring investors i(t) sin k p (t) inflation rate p.5 constant selling price p(t) sin( 2π k l t) variable selling price κ.8 rate of maximal borrowing ω 2. constant labor cost ω(t) 2. exp(.2 t) increasing labor cost δ.44 depreciation rate τ.5 tax rate σ.1 storage charges Table 1 Parameter and function values for the microeconomic control model. The economic process is considered on a time interval t [, t f ] with fixed time horizon t f >. The control constraints are given as box constraints, S c,min S c (t) S c,max, L c,min L c (t) L c,max, Y c,min Y c (t) Y c,max, I min I(t) I max, (6) for all t [, t f ], where we choose the following data: S c,min = 1, S c,max = 1, L c,min = 1, L c,max = 1, Y c,min = 1, Y c,max = 1, I min = 1, I max = 1. 4

5 The control constraints are written as u(t) U, where the cube U IR 4 is defined in an obvious way. The state inequality constraints are S min = 5 S(t), Y (t) κx(t), for all t [, t f ]. (7) The second state constraint imposes a maximal borrowing of loan capital. The further obvious state constraints x i (t) (i = 2,..., 6) do not become active and will therefore be omitted in the analysis of necessary conditions. Then the optimal control problem consists in determining a piecewise continuous (measurable) control u : [, t f ] IR 4 and an absolutely continuous state trajectory x : [, t f ] IR 6 that maximize the cost functional in Mayer form representing the joint capital of capital owners Φ(x(t f ), t f ) := X(t f ) + X m (t f ) + (1 τ) p(t f) d(t f ) S(t f) (8) subject to the constraints (5) (7). 3 Necessary optimality conditions A survey on necessary and sufficient conditions for state constrained optimal control problems may be found in Hartl, Sethi, Vickson [7]. The optimal control (5) (8) has the form of the control problem in Section 2 of [7], where the mixed control-state constraint is given by the simple box constraint (6). The state constraints (7) are written as h 1 (x) := S S min, h 2 (x) := κx Y. (9) We choose the direct adjoining approach described in Section 4 of [7], in which the state constraints (9) are directly adjoined to the Hamiltonian. Necessary conditions require regularity conditions for the state constraints which are associated with the order of a state constraint; cf. Section 2 in [7]. Both state constraints in (9) have order one, since h 1 1(x, u, t) = ḣ1 = Ṡ = S c, (1) h 1 2 (x, u, t) = ḣ2 = κẋ Ẏ = κ [ (I + (1 τ)(g(x, u, t) ρ r(t)x) ] Y c. In view of the gain function (2) we obtain h 1 1 u (x, u, t) = (1,,, ), h 1 2 (x, u, t) = (κ(1 τ)p(t),, 1, κ), (11) u d(t) 5

6 which implies that the regularity condition (2.11) in [7] holds, rank h1 1 / u = 2 h 1 2/ u (x, u, t). In particular, this holds along any boundary arc with h 1 (x(t)) = or h 2 (x(t)) = for t [t en, t ex ], where t en, resp., t ex denotes the entry-time, resp., the exittime of the boundary arc. On a boundary arc with h 1 (x(t)) =, the boundary control is given by h 1 1(x(t), u(t), t) = S c (t) for t [t en, t ex ]. (12) However, the boundary control on a boundary arc with h 2 (x(t)) = is not uniquely defined by the relation h 1 2(x, u, t) = h 1 2(x, I, Y c, t) =. On such a boundary arc, further relations defining the controls Y c and I can be obtained from the following necessary conditions of the Maximum Principle which is stated as Informal Theorem 4.1 in [7]. Since all control variables appear linearly in the control systems, the Hamiltonian can be written as H(x, u,, λ, t) = λf(x, u, t) = σ Sc S c + σ Lc L c + σ Yc Y c + σ I I (13) + R(x, λ, t), R(x, λ, t) = λ X (1 τ) ( p(t)α(x + Y ) α K L α L σs ω(t)l )/d(t) (14) λ X (1 τ)(ρ K (t)y + δ(x + Y ) + ρ r (t)x ) +λ Xm (1 τ)ρ m (t)x m + λ Xr (1 τ)ρ r (t)x. The adjoint variable λ = (λ S, λ L, λ Y, λ X, λ Xm, λ Xr ) IR 6 is a row-vector. The factors to the control components in the Hamiltonian are the switching functions σ Sc = λ S λ X (1 τ) p(t) d(t), σ L c = λ L, σ Yc = λ Y, σ I = λ X λ Xm. (15) Note that the switching vector σ = (σ Sc, σ Lc, σ Yc, σ I ) does not depend on the state variable x but only on the adjoint variable λ. This property will be important for the verification of sufficient conditions in Section 5. The Lagrangian is defined by adjoining the state constraints (9) directly to the Hamiltonian by a multiplier ν = (ν 1, ν 2 ) IR 2, L(x, u, λ, ν, t) = H(x, u, λ, t) + ν 1 (S 5) + ν 2 (κy X). (16) 6

7 For the economic control problem under investigation, the Maximum Principle can be rigorously justified using the techniques in Maurer [12]. Then the Informal Theorem 4.1 in Hartl et al [7] gives the following necessary conditions. Let (x(t), u(t)) be an optimal pair for the control problem (5) (8). For convenience, we drop asterisks or hats to denote an optimal solution. The notation [t] will be used to abbreviate arguments (x(t), u(t), λ(t), t). Then there exist a constant multiplier λ, a piecewise absolutely continuous adjoint function λ : [, t f ] IR 6, a piecewise continuous multiplier function ν : [, t f ] IR 2, a vector η(τ) = (η 1 (τ), η 2 (τ)) IR 2 for each junction time τ with a boundary arc, and a multiplier γ = (γ 1, γ 2 ) IR 2 with (λ, λ(t), ν(t), η(τ), γ) such that the following conditions hold for a.e. t [, t f ]: the minimum condition u(t) = arg max u U H(x(t), u, λ(t), t), (17) the adjoint equation λ(t) = L x [t], (18) the jump condition at a junction time λ(τ ) = λ(τ ) + η(τ)h x [τ], (19) the transversality condition at the terminal time λ(t f ) = λ Φ x [t f ] + γ 1 (h 1 ) x [t f ] + γ 2 (h 2 ) x [t f ], (2) and the complementarity conditions ν(t), ν(t)h[t] =, γ i, γ i h i [t f ] = (i = 1, 2). (21) The evaluation of the adjoint equations on the basis of equations (13), (14) and (16) is left to the reader. The transversality condition (2) yields in view of the cost functional (8) and the state constraint (9): λ S (t f ) = λ (1 τ)p(t f )/d(t f ) + γ 1, λ L (t f ) =, λ Y (t f ) = γ 2, λ X (t f ) = λ + γ 2 κ, λ Xm (t f ) = λ, λ Xr (t f ) =. (22) The numerical results in the next section show that the normality condition λ = 1 holds and the multipliers γ 1, γ 2 are zero, though both state constraints are active at the terminal time. The maximum condition (17) for the control 7

8 yields the following switching law for the control vector u = (u 1, u 2, u 3, u 4 ), where the switching functions σ i (t), i = 1, 2, 3, 4, are defined in (15): u i,max, if σ i (t) > u i (t) = u i,min, if σ i (t) < (23) singular, if σ i (t) = on [t en, t ex ] [, t f ] Here, t en, resp. t ex means the entry-time, resp., exit-time of a singular arc. On interior arcs with h(x(t)) <, one obtains further information on a singular control u i by differentiating the switching relation σ i (t) =, t [t en, t ex ]. We refrain from discussing this procedure in detail. On a boundary arc, the following property is noteworthy. If the component u i (t) of a boundary control lies in the interior of its control region, i.e., satisfies u i,min < u i (t) < u i,max for t en < t < t ex, then the maximum condition (17) implies σ i (t) = for t en < t < t ex. (24) Hence, the boundary control u i (t) formally behaves as a singular control. This property has been exploited in Maurer [11] to derive junction conditions for junctions between interior arcs and boundary arcs. Though the proof techniques in [11] have been developed only in the case of a scalar control, an inspection of the economic problem reveals that some junction results in [11] can be extended to the vector valued control case considered here. In particular, it follows that the adjoint variables are continuous on [, t f ], since the state constraints are of order one and the relevant control components are discontinuous at the entry times of the boundary arcs; cf. Corollary 5.2 (ii) and Theorem 5.4 in [11]. Thus the multipliers η(τ) in the jump condition (19) vanish. 4 Numerical solution and verification of necessary conditions The optimal control and state trajectory will be determined in two steps by a hybrid numerical approach where two direct optimization methods are combined. In the first step, the optimal control problem is discretized on a fixed grid. This leads to a large-scale opimization problem that may be solved by various nonlinear programming techniques. We have used two nonlinear programming implementations. Method-1: the control package NUDOCCCS by Büskens [3,4] with up to N = 1 grid points and a 4th order Runge Kutta 8

9 Fig. 1. Left: interest rates ρ K (t) (above) and ρ m (t)+ρ r (t); Right: inflation rate i(t). integration scheme; Method-2: the programming language AMPL [6] and the Interior Point Method code IPOPT developed by Wächter, Biegler [15] using up to N = 5. grid points with an EULER or HEUN integration scheme. Both methods are capable of detecting the correct control switching structure. In addition, Method (2) provides rather accurate estimates for the switching and junction times between bang bang and singular or boundary arcs. Moreover, Lagrange multipliers of the nonlinear programming problems can be identified with the values at grid points of the adjoint variables and multipliers for the state constraints. The second step is a refinement step, where the switching and junction times are determined with higher accuracy. Rather then optimizing the junction times directly, the arclengths of bang bang or singular arcs are treated as additional optimization variables. The implementation relies on a time-scaling and multiprocess control technique described in Büskens, Pesch, Winderl [5]. A simplified approach avoiding the multiprocess formulation may be found in Maurer, Büskens, Kim, Kaya [13]. Both methods and the refinement step have been tested in the diploma theses of Balzer [1] and Lang [9]. Now we present optimal control solutions for two data sets. The interest rates ρ K (t), ρ m (t) + ρ r (t) and the inflation rate i(t) given in Table 1 are depicted in Figure Solution for constant price p =.5, δ =.44 and constant wage ω = 2 Let us denote this data set by Data-1. The computed control u = (S c, L c, Y c, I) has a rather complicated switching structure with 8 bang bang and singular subarcs; cf. Table 2 with obvious notations. We obtain the following switching and junction times: t 1 = t 2 =.5, t 3 =.5666 t 4 = t 5 = t 6 = t 7 =

10 t S c L c Y c I 5 S.8X Y [., t 1 ] min max min min non-active non-active [t 1, t 2 ] min max max min non-active non-active [t 2, t 3 ] max max min active nonactive [t 3, t 4 ] min max min active non-active [t 4, t 5 ] min singular min active non-active [t 5, t 6 ] min singular singular active active [t 6, t 7 ] singular singular singular active active [t 7, t f ] max singular singular active active Table 2 Data-1: structure of optimal control with bang-bang, singular and boundary arcs Fig. 2. Stock S(t), employment L(t) and loan capital Y (t). The optimal functional value is Φ(x(t f )) = The optimal state trajectories x(t) = (S(t), L(t), Y (t), X(t), X m (t), X r (t)) are shown in Figs. 2 and 3. The control u(t) and the switching functions σ(t) are displayed in Figs The adjoint variable λ L and λ Y are shown in Figures 5, 6, while the adjoints λ S, λ X, λ Xm and the multiplier ν 2 for the state constraint κy X are displayed in Figs. 8, 9. The adjoint variable λ Xr vanishes identically. The computed initial values of the adjoints are λ S () = , λ L () = , λ Y () = , λ X () = , λ Xm () = , λ Xr () =.. The adjoint variables are continuous in [, t f ]. Hence, the multiplier η(τ) in the jump condition (19) vanishes. The terminal value is λ(t f ) = ( ,.,., 1., 1.,.) IR 6, which shows that the transversality condition (2) is satisfied with the multiplier λ = 1 and multipliers γ 1 = γ 2 =. The behavior of the switching functions (15) is in perfect agreement with the 1

11 Fig. 3. Equity capital X(t), alternative assets X m (t) and risk premium X r (t) Fig. 4. Stock control S c (t) and switching function σ Sc (t) Fig. 5. Employment control L c (t) and switching function σ Lc (t) = λ L (t) Fig. 6. Loan capital control Y c (t) and switching function σ Yc (t) = λ Y (t). control law (23) and the control structure in Table 2, since we have <, t < t 2 σ Sc (t) =, t 2 t 1, σ <, t < t 5 I(t) =, t 5 t t, f >, t < t 3 <, t < t 1 <, t 3 < t < t 6 σ Lc (t), σ Yc (t) >, t 1 < t < t 4. =, t 6 t t 7 =, t 4 t t f >, t 7 t t 7 11

12 Fig. 7. Investment control I(t) and switching function σ I (t) Fig. 8. Adjoint variables λ S (t), λ X (t) Fig. 9. Adjoint λ Xm (t) and multiplier ν 2 (t) for constraint.8 Y (t) X(t) Fig. 1. Zoom into the switching functions σ Lc (t) and σ Yc (t). In addition, the switching functions satisfy the so-called strict bang-bang property. Namely, σ k (t j ) holds at any switching time t j between bang-bang arcs of the control component u k. Figure 1 zooms into the switching function σ Lc (t) and σ Yc (t) to demonstrate in greater detail that (a) the control L c (t) has a junction of a singular arc with a bang-bang arc at t 7 = , and (b) the control Y c (t) switches between bang-bang arcs at t 1 = and has a junction with a singular arc at t 4 =

13 t S c L c Y c I 5 S.8X Y [., t 1 ] min max min max non-active non-active [t 1, t 2 ] min max max min non-active non-active [t 2, t 3 ] min max singular singular non-active active [t 3, t 4 ] max singular singular active active [t 4, t 5 ] min singular singular active active [t 5, t f ] max singular singular active active Table 3 Data-2 : structure of optimal control with bang-bang, singular and boundary arcs Fig. 11. Data-2 : Stock S(t), employment L(t) and loan capital Y (t). 4.2 Solution for variable price p(t) = sin(2πt/8), depreciation rate δ =.322 and increasing wage ω(t) = 2 exp(.2 t) We choose a data set denoted by Data-2 which is significantly different from that in Section 4.1 by considering the variable price p(t) =.5+.1 sin(2πt/8), depreciation rate δ =.322 and increasing wage ω(t) = 2 exp(.2 t); cf. [1]. Then the computed control u = (S c, L c, Y c, I) is a combination of 6 bang bang and singular subarcs that are described in Table 3. We obtain the switching and junction times t 1 =.1661, t 2 =.3842, t 3 =.5, t 4 = , t 5 = and the optimal functional value Φ(x(t f )) = The optimal state trajectories x(t) = (S(t), L(t), Y (t), X(t), X m (t), X r (t)) are shown in Figs. 11 and 12, while Figs depict the optimal control components jointly with the associated switching functions. The adjoint variables λ IR 6 and the multiplier ν 2 for the state constraint κy X are displayed in Figs. 14, 15, 17, 18. Again, we have λ Xr (t) 13

14 Fig. 12. Equity capital X(t), alternative assets X m (t) and risk premium X r (t). 5e e-5-1e Fig. 13. Stock control S c (t) and switching function σ Sc (t) Fig. 14. Employment control L c (t) and switching function σ Lc (t) = λ L (t) Fig. 15. loan capital control Y c (t) and switching function σ Yc (t) = λ Y (t). in [, t f ]. The computed initial values of the adjoints are λ S () = , λ L () = , λ Y () = , λ X () = , λ Xm () = , λ Xr () =., while the terminal value is λ(t f ) = ( ,.,., 1., 1.,.) IR 6. The switching functions (15) obey the control laws (23), resp., the control 14

15 Fig. 16. Investment control I(t) and switching function σ I (t) Fig. 17. adjoint variables λ S (t), λ X (t) Fig. 18. Adjoint variable λ Xm (t) and multiplier ν 2 (t) for constraint.8y (t) X(t). structure in Table 3 and also satisfy the strict bang-bang property at switching times between bang-bang arcs. 5 Verification of sufficient optimality conditions We are going to show that the sufficient optimality conditions of Arrow type in Hartl, Sethi, Vickson [7], Theorem 8.2, hold for both controls presented in section 4.1 and 4.2. The first assumption in Theorem 8.2 of [7] requires that the necessary conditions be satisfied with a multiplier λ = 1 in the transversality condition (22). This property holds as stated earlier in Section 4; cf. Figs. 5, 6, 8, 9 and Figs. 14, 15, 17, 18. The function Φ(x, t f ) = X + X m + S(1 τ)p(t f )/d(t f ) 15

16 defining the cost functional (8) is linear in x and, hence, concave in x. Note that we did not treat the discount rate d(t) defined by equation (3) as an auxiliary state variable x 7 as it was done in [8,16]. This additional state variable would destroy the concavity of the function Φ(x, t f ). The crucial condition then is the property that the maximized Hamiltonian H (x, λ, t) = max u U H(x, u, λ, t) is concave for all (λ(t), t), t [, t f ]. It can readily be seen from (13) and (14) that the maximized Hamiltonian is given by H (x, λ(t), t) = λ X (t)(1 τ) p(t)f (x) + R (t), (25) where F (x) = F (L, Y, X) = α(x + Y ) α K L α L is the Cobb Douglas production function (1) and R (t) does not depend on the state variable x. The production function F (x) = F (L, Y, X) is concave for L >, Y >, X > in view of the assumption < α, α K, α L and α K + α L < 1. This follows from the fact that the Hessian DxxF 2 (x) is negative semi definite since four eigenvalues are zero and two eigenvalues are negative. Moreover, Figs. 8 and 17 show that λ X (t) > holds for t [, t f ]. Since (1 τ)p(t) >, we finally conclude that the maxized Hamiltonian H (x, λ(t), t) in (25) is concave in x, which confirms that the computed controls are optimal. 6 Comparison and Conclusion The complicated control structure for Data-1 shows that many switchings between bang-bang arcs occur in the very beginning of the planning period, more precisely for t t 6 =.671. The reason for such multiple switchings may be that the initial values of loan and equity capital are not chosen properly. For the largest part of the planning period, namely for t 6 =.671 t t 7 = 8.94, all controls are singular and take values in the interior of the control region. Towards the end of the planning period, the employment rate L c (t) is increasing and takes its maximum value on the final arc [t 7, t f ]. The optimal solution for Data-2 is significantly different from the Data-1 solution as can be clearly seen in the behavior of the employment control L c (t). This is due to the variable price function p(t) which is increasing in the periods [, 2] and [6, 1], but decreasing in the period [2, 6]. The employment control L c (t) is bang-bang and takes its minimum negative value L c,min = 1, i.e. adopts a maximal dismissal rate, in the period [t 4, t 5 ] = [2.85, 6.2] before it switches to maximal hiring in the remaining planning period. When the price p(t) is decresaing, the loan capital control Y c (t) and the investment control I(t) have significantly smaller values than those for Data-1. 16

17 Due to the complexity of the control model, the complicated control structure can not be determined from a detailed discussion of the Maximum Principle alone. We have presented a hybrid numerical approach consisting of two consecutive direct optimization methods which yield control and state variables as well as junction times between bang-bang and singular arcs. Moreover, adjoint variables and multipliers associated with state constraints can be identified with Lagrange multipliers of the optimization problems. This allows us to verify necessary optimality conditions a posteriori. Using this approach, optimal solutions can be computed for various other data scenarios in the microeconomic control problem, e.g., for the risk premium rate ρ low r (t) for daring investors, etc. In all cases, the computed solution satisfies sufficient optimality conditions, since the maximized Hamiltonian turns out to be a concave function of the state variable. Acknowledgement: We are grateful to Nadja Balzer [1] and Matthias Lang [9] for numerical assistance with solving the control problems in Section 4.. References [1] Balzer, N., Optimale Steuerung ökonmischer Prozesse am Beispiel eines komplexen Unternehmensmodells mit bang bang, singulären Steueerungen und Zustandsbeschränkungen, Diploma Thesis, Institut für Numerische und Angewandte Mathematik, Universität Münster, 26. [2] Büskens, C., Direkte Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer und Zustands Beschränkungen. Dissertation, Institut für Numerische und Angewandte Mathematik, Universität Münster, [3] Büskens, C., and Maurer, H., SQP methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real time control, J. of Computational and Applied Mathematics 12, (2). [4] Büskens, C., and Maurer, H., Sensitivity analysis and real time control of parametric optimal control problems using nonlinear programming methods, in: Online Optimization of Large Scale Systems (Grötschel, M., Krumke, S.O., and Rambau, J., eds.), pp , Springer Verlag, Berlin, 21. [5] Büskens, C., Pesch, H.J., and Winderl, S., Real time solutions of bang bang and singular optimal control problems, in: Online Optimization of Large Scale Systems (Grötschel, M., Krumke, S.O., and Rambau, J., eds.), pp , Springer Verlag, Berlin, 21. [6] Fourer, R., Gay, D.M., and Kernighan, B.W., AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks/Cole Publishing Company,

18 [7] Hartl, R.F., Sethi, S.P., and Vickson, R.G., A survey of the maximum principles for optimal control problems with state constraints, SIAM Review 37, (1995). [8] Koslik, B., and Breitner, M.H., An optimal control problem in economics with four linear controls, J. Optimization Theory and Applications 94, (1997). [9] Lang, M., Schaltstrukturerkennung und Schaltpunkt Optimierung bei Optimalsteuerungsproblemen mit linear eingehenden Steuerungen am Beispiel eines Steuerungsproblems aus der Mikroökonomie, Diploma Thesis, Fakultät für Mathematik und Physik, Universität Bayreuth, 26. [1] Lesourne, J., and Leban, R., La substitution capital travail au cours de la croissance de l entreprise, Revue d Economie Politique 4, (1978). [11] Maurer, H., On optimal control problems with bounded state variables and control appearing linearly, SIAM J. Control and Optimization 15, (1977). [12] Maurer, H., On the minimum principle for optimal control problems with state constraints, Schriftenreihe des Rechenzentrums, Nr. 41, Universität Münster, [13] Maurer, H., Büskens, C., Kim, J.-H.R., Kaya, Y.C., Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control Methods and Applications 26, (25). [14] Maurer, H., Kim, J.-H.R., and Vossen, G., On a state constrained control problem in optimal production and maintenance, in: Optimal Control and Dynamic Games, Applications in Finance, Management Science and Economics, Deissenberg, C. and Hartl, R.F., eds., pp , Springer Verlag, 25. [15] Wächter, A., and Biegler, L.T., On the implementation of an interior point filterline search algorithm for large scale nonlinear programming, Mathematical Programming 16, (26). [16] Winderl, S, and Naumer, B., On a state constrained control problem in economics with four linear controls, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft Echtzeitoptimierung Großer Systeme, Report 9, 2. 18