Stabilized SQP revisited Izmailov, A. F.; Solodov, M. V.
Mathematical programming,
06/2012, Volume:
133, Issue:
1-2
Journal Article
Peer reviewed
Open access
The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated ...to a solution are not unique. Within the framework of Fischer (Math Program 94:91–124, 2002), the key to local superlinear convergence of sSQP are the following two properties: upper Lipschitzian behavior of solutions of the Karush-Kuhn-Tucker (KKT) system under canonical perturbations and local solvability of sSQP subproblems with the associated primal-dual step being of the order of the distance from the current iterate to the solution set of the unperturbed KKT system. According to Fernández and Solodov (Math Program 125:47–73, 2010), both of these properties are ensured by the second-order sufficient optimality condition (SOSC) without any constraint qualification assumptions. In this paper, we state precise relationships between the upper Lipschitzian property of solutions of KKT systems, error bounds for KKT systems, the notion of critical Lagrange multipliers (a subclass of multipliers that violate SOSC in a very special way), the second-order necessary condition for optimality, and solvability of sSQP subproblems. Moreover, for the problem with equality constraints only, we prove superlinear convergence of sSQP under the assumption that the dual starting point is close to a noncritical multiplier. Since noncritical multipliers include all those satisfying SOSC but are not limited to them, we believe this gives the first superlinear convergence result for any Newtonian method for constrained optimization under assumptions that do not include any constraint qualifications and are weaker than SOSC. In the general case when inequality constraints are present, we show that such a relaxation of assumptions is not possible. We also consider applying sSQP to the problem where inequality constraints are reformulated into equalities using slack variables, and discuss the assumptions needed for convergence in this approach. We conclude with consequences for local regularization methods proposed in (Izmailov and Solodov SIAM J Optim 16:210–228, 2004; Wright SIAM J. Optim. 15:673–676, 2005). In particular, we show that these methods are still locally superlinearly convergent under the noncritical multiplier assumption, weaker than SOSC employed originally.
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CEKLJ, DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, ODKLJ, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
The interaction of positively charged N‐terminal histone tails with nucleosomal DNA plays an important role in chromatin assembly and regulation, modulating their susceptibility to post‐translational ...modifications and recognition by chromatin‐binding proteins. Here, we report residue‐specific 15N NMR relaxation rates for histone H4 tails in reconstituted nucleosomes. These data indicate that H4 tails are strongly dynamically disordered, albeit with reduced conformational flexibility compared to a free peptide with the same sequence. Remarkably, the NMR observables were successfully reproduced in a 2‐μs MD trajectory of the nucleosome. This is an important step toward resolving an apparent inconsistency where prior simulations were generally at odds with experimental evidence on conformational dynamics of histone tails. Our findings indicate that histone H4 tails engage in a fuzzy interaction with nucleosomal DNA, underpinned by a variable pattern of short‐lived salt bridges and hydrogen bonds, which persists at low ionic strength (0–100 mM NaCl).
15N relaxation data suggest that amino‐terminal histone H4 tails in reconstituted nucleosome are flexible. This result is rationalized by suitably designed MD simulations, showing that H4 tails are involved in a fuzzy interaction with nucleosomal DNA while retaining their disordered character.
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
In the case of singular (and possibly even nonisolated) solutions of nonlinear equations, while superlinear convergence of the Newton method cannot be guaranteed, local linear convergence from large ...domains of starting points still holds under certain reasonable assumptions. We consider a linesearch globalization of the Newton method, combined with extrapolation and over-relaxation accelerating techniques, aiming at a speed up of convergence to critical solutions (a certain class of singular solutions). Numerical results indicate that an acceleration is observed indeed.
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CEKLJ, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
For the sequential quadratic programming method (SQP), we show that close to a solution satisfying the same assumptions that are required for its local quadratic convergence (namely uniqueness of the ...Lagrange multipliers and the second-order sufficient optimality condition), the direction given by the SQP subproblem using the Hessian of the Lagrangian is a descent direction for the standard
-penalty function. We emphasize that this property is not straightforward at all, because the Hessian of the Lagrangian need not be positive definite under these assumptions or, in fact, under any other reasonable set of assumptions. In particular, this descent property was not known previously, under any assumptions (even including the stronger linear independence constraint qualification, strict complementarity, etc.). We also check the property in question by experiments on nonconvex problems from the Hock-Schittkowski test collection for a model algorithm. While to propose any new and complete SQP algorithm is not our goal here, our experiments confirm that the descent condition, and a model method based on it, work as expected. This indicates that the new theoretical findings that we report might be useful for full/practical SQP implementations which employ second derivatives and linesearch for the
-penalty function. In particular, our results imply that in SQP methods where using subproblems without Hessian modifications is an option, this option has a solid theoretical justification at least on late iterations.
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BFBNIB, DOBA, GIS, IJS, IZUM, KILJ, KISLJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
This work continues a theoretical investigation of the capabilities of the well-known method based on using a monochromatic probe light beam in combination with optical pumping of atoms (molecules) ...of a rarefied-gas medium by a broadband radiation in a thin cell the diameter of which is much larger than its internal thickness. In contrast to calculations carried out in the previous publications on this method of spectroscopy, here, we consider the case of arbitrary values of pump intensity and thickness of a cylindrical gas cell. Thus, all the possible mechanisms and specificities of velocity selection of atoms in optically excited levels caused by transit-time relaxation of such atoms in gas cells of this kind are analyzed. Within the framework of this approach, sub-Doppler absorption resonances of the probe light beam corresponding to quantum transitions from the upper level excited by optical pumping are investigated. The obtained results can be used in high-resolution spectroscopy of atoms (molecules), as well as for laser-frequency stabilization to established narrow spectral resonances.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, SIK, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
It has been recently argued that in semi-classical gravity, a minimal 2-sphere is not a horizon but a tiny throat of a wormhole, such as the Damour–Solodukhin wormhole (DSWH), with a free parameter ...λ≠0 separating it from a Schwarxzschild black hole (BH) (λ=0). As shown by DS, their horizonless WH can mimic many properties of a black hole (BH). Assuming that observing a BH mimicker is equivalent to observing a BH itself, we ask the question as to which identity of the object, a WH or a BH, an observer is likely to observe in a single experiment. To answer this, we introduce Tangherlini’s new concept of indeterminacy in the gravitational field by portraying the field as a refractive medium. We then postulate that the identity of the observed object will depend on the probabilistic outcome of photon motion probing the object. The probabilities will be described by Fresnel reflection (R) and transmission (T) coefficients derived by Tangherlini on the basis of a non-quantum statistical indeterminacy of photon motion in ordinary optical media. By adapting this approach to a gravitational “effective optical medium,” we obtain two intriguing results: (i) The Fresnel coefficients at the DSWH throat are independent of mass M but dependent solely on the parameter λ≠0. (ii) Depending on the location of the observer, what is a DSWH to one observer may appear as a BH to another observer for the same value of λ≠0.
Abstract
Screening of the Protein Data Bank led to identification of a recurring structural motif where lysine NH
3
+
group interacts with backbone carbonyl. This interaction is characterized by ...linear atom arrangement, with carbonyl O atom positioned on the three-fold symmetry axis of the NH
3
+
group (angle C
ε
-N
ζ
-O close to 180°, distance N
ζ
-O ca. 2.7-3.0 Å). Typically, this linear arrangement coexists with three regular hydrogen bonds formed by lysine NH
3
+
group (angle C
ε
-N
ζ
-
acceptor atom
close to 109°, distance N
ζ
-
acceptor atom
ca. 2.7-3.0 Å). Our DFT calculations using polarizable continuum environment suggest that this newly identified linear interaction makes an appreciable contribution to protein’s energy balance, up to 2 kcal/mol. In the context of protein structure, linear interactions play a role in capping the C-termini of α-helices and 3
10
-helices. Of note, linear interaction involving conserved lysine is consistently found in the P-loop of numerous NTPase domains, where it stabilizes the substrate-binding conformation of the P-loop. Linear interaction NH
3
+
– carbonyl represents an interesting example of ion-dipole interactions that has so far received little attention compared to ion-ion interactions (salt bridges) and dipole-dipole interactions (hydrogen bonds), but nevertheless represents a distinctive element of protein architecture.
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IZUM, KILJ, NUK, PILJ, PNG, SAZU, UL, UM, UPUK
Immunotherapy in oncologic diseases involves the use of drugs which stimulate the immune system and indirectly suppress tumor cells growth. These agents have expanded the treatment options for cancer ...patients. Despite the impressive success achieved in the development of immune checkpoint inhibitors (ICIs) and subsequent approval in a broader spectrum of malignant tumors, most patients are not responded the therapy. Currently available predictive markers of efficacy are nonspecific. However, microRNAs are of particular interest, which regulate gene expression and are involved in the carcinogenesis and therapy resistance. Therefore, it is clear that for the most efficient and cost-effective use of ICIs, it is important to have validated biomarkers that will accurately predict the therapeutic response. The published results on molecular genetic changes in patients with renal cell carcinoma (RCC) were analyzed and summarized in order to determine possible prognostic biomarkers when prescribing ICI therapy.
•We revealed trap energies and densities in FE and non-FE HfO2 and HfZrO4 layers.•The high deep trap density (~1019 cm−3) poses challenges for electrostatic control.•The observed traps are likely ...intrinsic to hafnia/zirconia lattice (polarons).
Electron trapping in ferroelectric and non-ferroelectric HfO2-based layers was studied at room temperature by charge injection and photodepopulation techniques. The comparison of inferred energy distribution and density of trapped electrons in differently processed samples shows insignificant impact of Al- and Si-doping on trapping properties suggesting intrinsic nature of the observed traps. A comparison to the HfZrO4 layers is provided. The volume concentration of deep traps, most of which are energetically distributed between 2 and 3.5 eV below the HfO2 conduction band, is found to be in the range of 1019 cm−3.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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