### Table 2 Terms of the convex optimization problem depending on the choice of the loss function.

"... In PAGE 16: ...Table 2 Terms of the convex optimization problem depending on the choice of the loss function. These two cases can be combined into 2 [0; C] and T ( ) = ?p ? 1 p C? p p?1 p p?1 : (48) Table2 contains a summary of the various conditions on and formulas for T ( ) for di erent cost functions.6 Note that the maximum slope of ~ c determines the region of feasibility of , i.... ..."

### 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 2: Overview of prior work on learning Markov networks that has formal guarantees. More details are given in Section 5.1. The references in the table are: [1]: Chow and Liu (1968); [2] Srebro (2001); [3]: Narasimhan and Bilmes (2004); [4]: Besag (1974b); [5]: Gidas (1988); [6]: this paper. Convex refers to the time of solving a convex optimization problem.

2006

Cited by 4

### Table 2: Overview of prior work on learning Markov networks that has formal guarantees. More details are given in Section 5.1. The references in the table are: [1]: Chow and Liu (1968); [2] Srebro (2001); [3]: Narasimhan and Bilmes (2004); [4]: Besag (1974b); [5]: Gidas (1988); [6]: this paper. Convex refers to the time of solving a convex optimization problem.

2006

Cited by 4

### 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

### Table 2. Problem dimensions. Recall that the dimension of each BMI is equal to m, hence depends on the order of the plant. One can see that the number of BMIs quickly becomes very large as the order of the controller increases. Considering controllers of higher orders, or with more design parameters than those proposed in Table 1, would have led to systems of 256 BMIs or more that must be discarded for obvious practical reasons. This is a main hindrance to the design of high-order controllers, and helps to pinpoint the origin of the di culty of robust stabilization of interval plants. In [17], the authors showed that many of the problems considered in the robust control literature can be formulated as BMIs. In view of Theorem 3, this is also the case for the robust stabilization of interval plants. Unfortunately, BMIs are highly non-convex optimization problems and solving a general BMI was shown to be NP-hard [29].

2001

"... In PAGE 7: ... Theorem 3 Any vector z solution to the BMIs n X j=0 n X k=0 zjzkHijk gt; 0; i = 1; : : : ; N (3) parametrizes a pair of polynomials x(s) and y(s) giving rise to a robustly stabilizing con- troller for interval plant (1). In Table2 , we report the number of BMIs that occur for di erent controller architectures, together with the number of decision variables, or design parameters.... ..."

Cited by 2

### Table 3: Uniform Model with the convex quadratic valid inequalities (21). n is the number of customer segments, m is the number of products, and v is a label of the problem instance. The column \MIP quot; is the optimal objective value (4), the column \LP quot; is the optimal objective value of the LP relaxation of (4), and the column \With Cut quot; is the optimal objective value of the continuous relaxation of (4) with the convex quadratic inequality (21).

2007

"... In PAGE 29: ... We see that these inequalities are indeed cuts since the optimal solution of the LP violates them in most instances. There are four anomalies in Table3 , namely, the instances (n; m; v) with (10, 40, 5), (10, 60, 1), (10, 60, 4) and (10, 60, 5). For each of these instances, the objective value of the QCP relaxation is strictly less than the optimal objective value of the MIP.... ..."

Cited by 1