Solving SDP completely with an interior point oracle

Lourenço, Bruno F.; Muramatsu, Masakazu; Tsuchiya, Takashi

doi:10.1080/10556788.2020.1850720

Cited by 5 publications

(3 citation statements)

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“…In this case, as described below, we can find a scaling such that the sum of eigenvalues of any feasible solution of P S ∞ (A) is bounded by r . Let H be the set of indices i satisfying (11) for each block . According to Proposition 3.5, set…”

Section: Proposition 36 Suppose That For Anymentioning

confidence: 99%

“…In general, it is difficult to solve such problems stably by using interior point methods, but since strong complementarity exists between (P) and (D), they can be expected to be stably solved. By applying Lemma 3.4 of [11], we can generate a problem in which both the primal and dual problems have feasible interior points in which it can be determined whether (P) has a feasible interior point. However, since there was no big difference between the solution obtained by solving the problem generated by applying Lemma 3.4 of [11] and the solution obtained by solving the above (P) and (D), we showed only the results of solving (P) and (D) above.…”

Section: Outline Of Numerical Implementationmentioning

confidence: 99%

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A new extension of Chubanov’s method to symmetric cones

Kanoh

Yoshise

2023

Math. Program.

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We propose a new variant of Chubanov’s method for solving the feasibility problem over the symmetric cone by extending Roos’s method (Optim Methods Softw 33(1):26–44, 2018) of solving the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound for the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (1) equivalent to that of Roos’s original method (2018) and superior to that of Lourenço et al.’s method (Math Program 173(1–2):117–149, 2019) when the symmetric cone is the nonnegative orthant, (2) superior to that of Lourenço et al.’s method (2019) when the symmetric cone is a Cartesian product of second-order cones, (3) equivalent to that of Lourenço et al.’s method (2019) when the symmetric cone is the simple positive semidefinite cone, and (4) superior to that of Pena and Soheili’s method (Math Program 166(1–2):87–111, 2017) for any simple symmetric cones under the feasibility assumption of the problem imposed in Pena and Soheili’s method (2017). We also conduct numerical experiments that compare the performance of our method with existing methods by generating strongly (but ill-conditioned) feasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.

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Section: Proposition 36 Suppose That For Anymentioning

confidence: 99%

Section: Outline Of Numerical Implementationmentioning

confidence: 99%

A new extension of Chubanov’s method to symmetric cones

Kanoh

Yoshise

2023

Math. Program.

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“…While many of the earlier papers on facial reduction focused on weakly feasible problems, it is relatively recent that weak infeasibility is analyzed in this context [15,18,29]. Along this line of developments, the paper [20] showed that, through double facial reduction, any SDP can be solved "completely" by calling an interiorpoint oracle polynomially many times, where the interior-point oracle is an idealized interior-point algorithm which return primal-dual optimal solutions given a primaldual strongly feasible SDP. In the context of SDPs with positive duality gaps, Ramana developed an extended Lagrangian dual SDP for which strong duality always holds [34].…”

Section: Introductionmentioning

confidence: 99%

A limiting analysis on regularization of singular SDP and its implication to infeasible interior-point algorithms

et al. 2022

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We consider primal-dual pairs of semidefinite programs and assume that they are singular, i.e., both primal and dual are either weakly feasible or weakly infeasible. Under such circumstances, strong duality may break down and the primal and dual might have a nonzero duality gap. Nevertheless, there are arbitrary small perturbations to the problem data which would make them strongly feasible thus zeroing the duality gap. In this paper, we conduct an asymptotic analysis of the optimal value as the perturbation for regularization is driven to zero. Specifically, we fix two positive definite matrices, $$I_p$$ I p and $$I_d$$ I d , say, (typically the identity matrices), and regularize the primal and dual problems by shifting their associated affine space by $$\eta I_p$$ η I p and $$\varepsilon I_d$$ ε I d , respectively, to recover interior feasibility of both problems, where $$\varepsilon $$ ε and $$\eta $$ η are positive numbers. Then we analyze the behavior of the optimal value of the regularized problem when the perturbation is reduced to zero keeping the ratio between $$\eta $$ η and $$\varepsilon $$ ε constant. A key feature of our analysis is that no further assumptions such as compactness or constraint qualifications are ever made. It will be shown that the optimal value of the perturbed problem converges to a value between the primal and dual optimal values of the original problems. Furthermore, the limiting optimal value changes “monotonically” from the primal optimal value to the dual optimal value as a function of $$\theta $$ θ , if we parametrize $$(\varepsilon , \eta )$$ ( ε , η ) as $$(\varepsilon , \eta )=t(\cos \theta , \sin \theta )$$ ( ε , η ) = t ( cos θ , sin θ ) and let $$t\rightarrow 0$$ t → 0 . Finally, the analysis leads us to the relatively surprising consequence that some representative infeasible interior-point algorithms for SDP generate sequences converging to a number between the primal and dual optimal values, even in the presence of a nonzero duality gap. Though this result is more of theoretical interest at this point, it might be of some value in the development of infeasible interior-point algorithms that can handle singular problems.

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