### 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 1: List of nonconvex and convex potential functions that have been used.

1998

"... In PAGE 6: ... Depending on the choice of the potential function, (2) includes many common MRF models that have been proposed in the literature. Table1 lists a variety of such potential functions. Notice that only the GGMRF model depends on p through the potential function.... In PAGE 10: ...4 ML Estimate of and p for Non-scalable Priors In this section, we derive methods to compute the joint ML estimates of and p when the potential function is not scalable. This includes all the potential functions of Table1 except the Gaussian, Laplacian, and GGMRF. Notice that u(x; p) is not a function of p for any of the non-scalable potential functions.... ..."

Cited by 34

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

2005

### 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.... ..."

Cited by 4

### TABLE I PERFORMANCE REACHED BY DC-HAEA AND DC ON THE TEST FUNCTIONS USING DIFFERENT ENCODING SCHEME.

### Table 1: Optimization results for the non-convex test problem. DN(5,3) denotes the discrete neighboorhood of (5,3).

### Table 1: Optimization results for the non-convex test problem. DN#285,3#29 denotes the discrete neighboorhood

in Multipoint

2000