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ESE 605-001: Modern Convex Optimization
创建日期：2024-07-09 15:09:38
所属分类：经济Economics
标签： ESE 605
Modern Convex Optimization

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ESE 605-001: Modern Convex Optimization


1.  Optimal  spacecraft  landing.  We consider the problem of optimizing the thrust proile for a spacecraft to carry out a landing at a target position. The spacecraft dynamics are

where m > 0 is the spacecraft mass, e3 = [0, 0, 1]T, p(t) 2 R3  is the spacecraft position, with 0 the target landing position and p3 (t) representing height, f(t) 2 R3  is the thrust force, and g > 0 is the gravitational acceleration.  (For simplicity we assume that the spacecraft mass is constant. This is not always a good assumption, since the mass decreases with fuel use.  We will also ignore any atmospheric friction.) We must have p(Ttd ) = 0 and˙(p)(Ttd ) = 0, where Ttd  is the touchdown time.  The spacecraft must remain in a region given by

where α > 0 is a given minimum glide slope.  The initial position p(0) and velocity ˙(p)(0) are given.

The thrust force f(t) is obtained from a single rocket engine on the spacecraft, with a given maximum thrust; an attitude control system rotates the spacecraft to achieve any desired direction of thrust. The thrust force is therefore characterized by the constraint  If(t)I2          Fmax.  The fuel use rate is proportional to the thrust force magnitude, so the total fuel use is

where    > 0 is the fuel consumption coefficient.  The thrust force is discretized in time,i.e., it is constant over consecutive time periods of length h > 0, with f(t) = fk  for t 2 [(k - 1)h, kh), for k = 1, . . . , K, where Ttd  = Kh.  Therefore we have vk+1  = vk  + (h/m)fk  - hge3 ,    pk+1  = pk  + (h/2)(vk  + vk+1), where pk  denotes p((k - 1)h), and vk  denotes˙(p)((k - 1)h). We will work with this discrete-time model.

For simplicity, we will impose the glide slope constraint only at the times t = 0, h, 2h, ..., Kh.

(a) Minimum fuel descent.   Explain how to ind the thrust proile  f1 , . . . , fK   that minimizes fuel consumption, given the touchdown time Ttd  = Kh and discretization time h.

(b) Minimum time descent.  Explain how to ind the thrust proile that minimizes the touch-down time, i.e., K, with hixed and given. Your method can involve solving several convex optimization problems.

(c)  Carry out the methods described in parts (a) and (b) above on the problem instance with data given in Prob1.mat.  Report the optimal total fuel consumption for part  (a), and the minimum touchdown time for part (b).  Helper functions for plotting code (commented out) can be found in Prob1.* to help you visualize your solution.  Use the code to plot the spacecraft trajectory and thrust proiles you obtained for parts (a) and (b).

2.  Maximizing  algebraic  connectivity  of a  graph.   Let  G  = (V, E) be a weighted undirected graph with n = jVj nodes, m = jEj edges, and weights w1 , . . . , wm  2 R+  on the edges. If edge k connects nodes i and j, then deine ak  2 Rn  as (ak )i  = 1, (ak )j  = — 1, with other entries zero.  The weighted Laplacian (matrix) of the graph is deined as

where A = [a1   · · ·  am ]  2 Rnxm  is the incidence matrix of the graph.  Nonnegativity of the weights implies L > 0.

Denote the eigenvalues of the Laplacian L as

which  are functions of w.   The  minimum  eigenvalue  λ1   is  always  zero,  while  the  second  smallest eigenvalue λ2  is called the algebraic  connectivity of G and is a measure of the connectedness of a graph: The larger λ2  is, the better connected the graph is.  It is often used, for example, in analyzing the robustness of computer networks.

Though not relevant for the rest of the problem, we mention a few other examples of how the algebraic connectivity can be used.  These results, which relate graph-theoretic properties of G to properties of the spectrum of L, belong to a ield called spectral graph  theory.  For example,  λ2  > 0 if and only if the graph is connected. The eigenvector v2  associated with λ2  is often called the Fiedler vector  and is widely used in a graph partitioning technique called spectral partitioning, which assigns nodes to one of two groups based on the sign of the relevant component in v2.  Finally, λ2  is also closely related to a quantity called the isoperimetric  number  or  Cheeger  constant  of G, which measures the degree to which a graph has a bottleneck.

The problem is to choose the edge weights w  2  Rm+subject to some linear inequalities (and the
 nonnegativity constraint) so as to maximize the algebraic connectivity:

with variable w 2 Rm .

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