### Table 11: Impact of non-convexity

2007

"... In PAGE 26: ...Table 11: Impact of non-convexity These cases are analyzed in Table11 , where, in percentages, \robust nominal quot; is the nominal return attained by the optimal solution to the robust optimization problem and \robust worst case quot; is the worst-case return it attains under the uncertainty model; \robust positions quot; is the number of positions taken by the robust portfolio. From an aggregate perspective all six cases are equivalent: the adversary can, in each case, decrease returns by a total \mass quot; of 100.... In PAGE 26: ... From an aggregate perspective all six cases are equivalent: the adversary can, in each case, decrease returns by a total \mass quot; of 100. Yet, as we can see from Table11 , the six cases are structurally quite di erent. It appears, therefore, that a smooth convex model used to replace our histogram structure would likely produce very di erent results in at least some of the six cases.... ..."

### Table 4 Results on solving the four 730-hour non-convex models and their convex Model Non-convex

2005

### Table 6: Commands for dealing with non-convex piecewise linear sets.

"... In PAGE 25: ... For convenience, we build a separate toolkit in order to deal with non-convex Pwl sets. Table6 lists the commands for operations on Pwl sets.... ..."

### Table 3: 16 non-convex links, ratio 6:1, 10000 frames.

2004

"... In PAGE 5: ... It is clear that OBB approximate the body geometry better than AABBs and less dependent on length to thickness ratio. Table3 is for non-convex bodies. Figure 3(c) shows the dented links reminiscent of a boat hull.... ..."

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### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

in Abstract

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convex optimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."

### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

in Abstract

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convex optimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."

### Table 3: TemPTo relations on non-convex time intervals

"... In PAGE 7: ...ote that a convex time interval can be seen as a non-convex time interval (i.e. a collection with one component). Table 2, Table3 , and Table 4 give the relation and functions o ered by the qualitative layer. The de nitions of these relations and functions are as usual (cf.... ..."

### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convexoptimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."

### Table 1: Results for Quadratic Non-Convex Functions Number of Number of Precision Time of

1999

"... In PAGE 14: ... We chose a xed initial value of p which was increased if the optimal solution was not found after a given number of iterations. In Table1 - Table 2 (Appendix I) the computational results for quadratic and nonsmooth functions are presented, respectively. The space dimension for the test examples varied between 2 and 20 variables.... ..."

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