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The study of defect centers in silicon has been recently reinvigorated by their potential applications in optical quantum information processing. A number of silicon defect centers emit single photons in the telecommunication $O$-band, making them promising building blocks for quantum networks between computing nodes. The two-carbon G-center, self-interstitial W-center, and spin-$1/2$ T-center are the most intensively studied silicon defect centers, yet despite this, there is no consensus on the precise configurations of defect atoms in these centers, and their electronic structures remain ambiguous. Here we employ \textit{ab initio} density functional theory to characterize these defect centers, providing insight into the relaxed structures, bandstructures, and photoluminescence spectra, which are compared to experimental results. Motivation is provided for how these properties are intimately related to the localization of electronic states in the defect centers. In particular, we present the calculation of the zero-field splitting for the excited triplet state of the G-center defect as the structure is transformed from the A-configuration to the B-configuration, showing a sudden increase in the magnitude of the $D_{zz}$ component of the zero-field splitting tensor. By performing projections onto the local orbital states of the defect, we analyze this transition in terms of the symmetry and bonding character of the G-center defect which sheds light on its potential application as a spin-photon interface., Comment: 11 pages, 6 figures

Quantum light sources play a fundamental role in quantum technologies ranging from quantum networking to quantum sensing and computation. The development of these technologies requires scalable platforms, and the recent discovery of quantum light sources in silicon represents an exciting and promising prospect for scalability. The usual process for creating color centers in silicon involves carbon implantation into silicon, followed by rapid thermal annealing. However, the dependence of critical optical properties, such as the inhomogeneous broadening, the density, and the signal-to-background ratio, on centers implantation steps is poorly understood. We investigate the role of rapid thermal annealing on the dynamic of the formation of single color centers in silicon. We find that the density and the inhomogeneous broadening greatly depend on the annealing time. We attribute the observations to nanoscale thermal processes occurring around single centers and leading to local strain fluctuations. Our experimental observation is supported by theoretical modeling based on first principles calculations. The results indicate that annealing is currently the main step limiting the scalable manufacturing of color centers in silicon.

${\mathrm{Co}}_{3}{\mathrm{Sn}}_{2}{\mathrm{S}}_{2}$ is a magnetic Weyl semimetal, in which ferromagnetic ordering at 177 K is predicted to stabilize Weyl points. We perform temperature and spatial dependent angle-resolved photoemission spectroscopy measurements through the Curie temperature (${T}_{c}$), which show large band shifts and renormalization concomitant with the onset of magnetism. We argue that ${\mathrm{Co}}_{3}{\mathrm{Sn}}_{2}{\mathrm{S}}_{2}$ evolves from a Mott ferromagnet below ${T}_{c}$ to a correlated metallic state above ${T}_{c}$. To understand the magnetism, we derive a tight-binding model of Co-$3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ orbitals on the kagome lattice. At the filling obtained by first-principles calculations, this model reproduces the ferromagnetic ground state, and results in the reduction of Coulomb interactions due to cluster effects. Using a disordered local moment simulation, we show how this reduced Hubbard $U$ leads to a collapse of the bands across the magnetic transition, resulting in a correlated state, which carries associated characteristic photoemission signatures that are distinct from those of a simple lifting of exchange splitting. The behavior of topology across ${T}_{c}$ is discussed in the context of this description of the magnetism.

This article explores the possibility of improving the accuracy of positioning in single-frequency satellite radio navigation equipment through the use of an empirical model of the total electronic content GEMTEC. The effectiveness of this model is compared with that of the Klobuchar model, which is recommended for the GPS interface control document. We conducted testing at our observation points, using data from the international network of IGS stations in the GPS system. The use of the international network allowed us to select a long period of time for the testing from 2001 to 2017. As a result, it was shown that the GEMTEC model significantly reduces the average positioning errors as compared to the Klobuchar model. We also demonstrate the possibility of introducing the GEMTEC model and its full-featured use in single-frequency home-class receivers, for example, in the Russian receiver MNP-M7.

Using the density-functional-based LDA+U method and linear-response theory, we study the magnetic exchange interactions of the superconductor Nd1-xSrxNiO2. Our calculated nearest-neighbor exchange constant J1=82 meV is large, weakly affected by doping, and is only slightly smaller than that found in the sister compound CaCuO2. However, we find that the hole doping significantly enhances the interlayer exchange coupling as it affects the magnetic moment of the Ni-3d3z2-r2 orbital. This can be understood in terms of the small hybridization of Ni-3d3z2-r2 within the NiO2 plane, which results in a flat band near the Fermi level, and its large overlap along the z direction. We also demonstrate that the Nd-5d states appearing at the Fermi level do not affect the magnetic exchange interactions, and thus they may not participate in the superconductivity of this compound. Whereas many previous works emphasized the importance of the Ni-3dx2-y2 and Nd-5d orbitals, we analyze instead the solution of the Ni-3dx2-y2/Ni-3d3z2-r2 minimal model using dynamical mean field theory. It reveals an underlying Mott insulating state that, depending on the precise values of the intra-atomic Hund's coupling smaller or larger than 0.83 eV, selects upon doping either S=0 or 1 two-hole states at low energies, leading to very different quasiparticle band structures. We propose that trends upon doping in the spin excitation spectrum and the quasiparticle density of states can be a way to probe the Ni 3d8 configuration. © 2021 American Physical Society. X.W. is supported by the NSFC (Grants No. 11834006, No. 11525417, No. 51721001, and No. 11790311), National Key R&D Program of China (Grants No. 2018YFA0305704 and No. 2017YFA0303203), and by 111 Project. X.W. also acknowledges the support from the Tencent Foundation through the XPLORER PRIZE. V.I., G.R., and S.Y.S. are supported by NSF DMR Grant No. 1832728. I.L. acknowledges support by the Russian Foundation for Basic Research (Project No. 18-32-20076). The DMFT electronic structure calculations were supported by the state assignment of Minobrnauki of Russia (theme “Electron” No. AAAA-A18-118020190098-5).

Non-centrosymmetric superconductors have recently attracted much attention, since the lack of inversion symmetry mixes spin singlet and triplet pairing states, which may allow the realization of topological superconductivity. In this work, we study the electronic properties of the family of inversion-broken CeTX$_3$ heavy-fermion superconductors, finding topological nodal lines as well as Dirac and Weyl points, which are renormalized closer to the Fermi energy by correlations. We find that the Weyl nodal lines have a substantial effect on the Fermi surface spin structure of the normal state, and lead to line nodes in the superconducting phase., Comment: 6 pages, 4 figures

Uranium-based ferromagnet demonstrates notable ability to convert heat to electricity., The transverse voltage generated by a temperature gradient in a perpendicularly applied magnetic field, termed the Nernst effect, has promise for thermoelectric applications and for probing electronic structure. In magnetic materials, an anomalous Nernst effect (ANE) is possible in a zero magnetic field. We report a colossal ANE in the ferromagnetic metal UCo0.8Ru0.2Al, reaching 23 microvolts per kelvin. Uranium’s 5f electrons provide strong electronic correlations that lead to narrow bands, a known route to producing a large thermoelectric response. In addition, uranium’s strong spin-orbit coupling produces an intrinsic transverse response in this material due to the Berry curvature associated with the relativistic electronic structure. Theoretical calculations show that in UCo0.8Ru0.2Al at least 148 Weyl nodes, and two nodal lines, exist within 60 millielectron volt of the Fermi level. This work demonstrates that magnetic actinide materials can host strong Nernst and Hall responses due to their combined correlated and topological nature.

Although strong electronic correlations are known to be responsible for some highly unusual behaviors of solids such as metal-insulator transitions, magnetism, and even high-temperature superconductivity, their interplay with recently discovered topological states of matter awaits a full exploration. Here, we use a modern electronic structure method, combining the density functional theory of band electrons with dynamical self-energies of strongly correlated states, to predict that two well-known phases of actinide compound UNiSn, a paramagnetic semiconducting and antiferromagnetic metallic, correspond to topological insulator (TI) and Weyl semimetal (WSM) phases of topological quantum matter. Thus, the famous unconventional insulator-metal transition observed in UNiSn is also a TI-to-WSM transition. Driven by a strong hybridization between U f-electron multiplet transitions and band electrons, multiple energy gaps open up in the single-particle spectrum whose topological physics is revealed using the calculation of Z_{2} invariants in the strongly correlated regime. A simplified physical picture of these phenomena is provided based on a periodic Anderson model of strong correlations and multiple band inversions that occur in this fascinating compound. Studying the topology of interacting electrons reveals interesting opportunities for finding exotic phase transitions in strongly correlated systems.

Although topological invariants have been introduced to classify the appearance of protected electronic states at surfaces of insulators, there are no corresponding indexes for Weyl semimetals whose nodal points may appear randomly in the bulk Brillouin Zone (BZ). Here we use a well-known result that every Weyl point acts as a Dirac monopole and generates integer Berry flux to search for the monopoles on rectangular BZ grids that are commonly employed in self-consistent electronic structure calculations. The method resembles data mining technology of computer science and is demonstrated on locating the Weyl points in known Weyl semimetals. It is subsequently used in high throughput screening several hundreds of compounds and predicting a dozen new materials hosting nodal Weyl points and/or lines., Comment: 6 pages, 4 figures, 1 table, supplementary info (1 table, 13 figures)

Single-walled carbon nanotubes are particularly attractive for biomedical applications, because they exhibit a fluorescent signal in a spectral region where there is minimal interference from biological media. Although single-walled carbon nanotubes have been used as highly sensitive detectors for various compounds, their use as in vivo biomarkers requires the simultaneous optimization of various parameters, including biocompatibility, molecular recognition, high fluorescence quantum efficiency and signal transduction. Here we show that a polyethylene glycol ligated copolymer stabilizes near-infrared-fluorescent single-walled carbon nanotubes sensors in solution, enabling intravenous injection into mice and the selective detection of local nitric oxide concentration with a detection limit of 1 µM. The half-life for liver retention is 4 h, with sensors clearing the lungs within 2 h after injection, thus avoiding a dominant route of in vivo nanotoxicology. After localization within the liver, it is possible to follow the transient inflammation using nitric oxide as a marker and signalling molecule. To this end, we also report a spatial-spectral imaging algorithm to deconvolute fluorescence intensity and spatial information from measurements. Finally, we demonstrate that alginate-encapsulated single-walled carbon nanotubes can function as implantable inflammation sensors for nitric oxide detection, with no intrinsic immune reactivity or other adverse response for more than 400 days., Arnold and Mabel Beckman Foundation National Institute of Environmental Health Sciences: P30 ES002109 National Institutes of Health: ES007020 National Cancer Institute: P01 CA26731 2211, This work was supported by the National Institutes of Health (T32 Training Grant in Environmental Toxicology ES007020, to N.I.), the National Cancer Institute (grant P01 CA26731), the National Institute of Environmental Health Sciences (grant P30 ES002109), a Beckman Young Investigator Award and a National Science Foundation Presidential Early Career Award for Scientists and Engineers (to M.S.S.), the TUBITAK 2211 and 2214 Research fellowship programme (F.S. and S.S.) and the METU-DPT-OYP programme (F.S. and S.S). A Biomedical Innovation grant from Sanofi Aventis to M.S.S. is also acknowledged. The authors thank T. Grusecki for assistance with the deconvolution code, as well as R. Langer, D. Anderson and P. Dedon for helpful discussions.

Implantable, near infrared (nIR) fluorescent nanosensors are advantageous for in vivo monitoring of biological analytes since they can be rendered selective for particular target molecule while utilizing their unique optical properties and the nIR tissue transparency window for information transfer without an internal power source or telemetry. However, basic questions remain regarding the optimal encapsulation platform, geometrical properties, and concentration ranges required for effective signal to noise ratio through biological tissue. In this work, we systematically explore these variables quantitatively to optimize the performance of such optical nanosensors for biomedical applications. We investigate both alginate and polyethylene glycol (PEG) as model hydrogel systems, encapsulating d(GT)[subscript 15] ssDNA-wrapped single walled carbon nanotubes (SWNT) as model fluorescent nanoparticle sensors, responsive to riboflavin. Hydrogel sensors implanted 0.5 mm into thick tissue samples cause 50% reduction of initial fluorescence intensity, allowing an optical detection limit of 5.4 mm and 5.1 mm depth in tissue for alginate and PEG gels, respectively, at a SWNT concentration of 10 mg L−1, and 785 nm laser excitation of 80 mW and 30 s exposure. These findings are supported with in vivo nIR fluorescent imaging of SWNT hydrogels implanted subcutaneously in mice. For the case of SWNT, we find that the alginate system is preferable in terms of emission intensity, sensor response, rheological properties, and shelf life., National Institutes of Health (U.S.) (T32 Training Grant in Environmental Toxicology ES007020), National Cancer Institute (U.S.) (Grant P01 CA26731), National Institute of Environmental Health Sciences (Grant P30 ES002109), Arnold and Mabel Beckman Foundation (Young Investigator Award), National Science Foundation (U.S.) (Presidential Early Career Award for Scientists and Engineers), MIT-Technion Fellowship, Samsung Scholarship Foundation, Sanofi Aventis (Firm) (Biomedical Innovation Grant)

A rare combination of strong spin-orbit coupling and electron-electron correlations makes the iridate Mott insulator Sr$_2$IrO$_4$ a promising host for novel electronic phases of matter. The resemblance of its crystallographic, magnetic and electronic structures to La$_2$CuO$_4$, as well as the emergence upon doping of a pseudogap region and a low temperature $d$-wave gap, has particularly strengthened analogies to cuprate high-$T_c$ superconductors. However, unlike the cuprate phase diagram that features a plethora of broken symmetry phases in a pseudogap region that include charge density wave, stripe, nematic and possibly intra-unit cell loop-current orders, no broken symmetry phases proximate to the parent antiferromagnetic Mott insulating phase in Sr$_2$IrO$_4$ have been observed to date, making the comparison of iridate to cuprate phenomenology incomplete. Using optical second harmonic generation, we report evidence of a hidden non-dipolar magnetic order in Sr$_2$IrO$_4$ that breaks both the spatial inversion and rotational symmetries of the underlying tetragonal lattice. Four distinct domain types corresponding to discrete 90$^{\circ}$ rotated orientations of a pseudovector order parameter are identified using nonlinear optical microscopy, which is expected from an electronic phase that possesses the symmetries of a magneto-electric loop-current order. The onset temperature of this phase is monotonically suppressed with bulk hole doping, albeit much more weakly than the N\'eel temperature, revealing an extended region of the phase diagram with purely hidden order. Driving this hidden phase to its quantum critical point may be a path to realizing superconductivity in Sr$_2$IrO$_4$., Comment: 13 pages main text, 4 figures, 21 pages supplementary information

We introduce and study three-dimensional quantum dimer models with positive resonance terms. We demonstrate that their ground state wave functions exhibit a nonlocal sign structure that can be exactly formulated in terms of loops, and as a direct consequence, monomer excitations obey Fermi statistics. The sign structure and Fermi statistics in these "signful" quantum dimer models can be naturally described by a parton construction, which becomes exact at the solvable point., Comment: 9 pages, 12 figures

In this paper, we prove that a radially lower semicontinuous function of one variable, defined on some interval, is pseudoconvex, if and only if its domain of definition can be split into three parts such that the function is strictly monotone decreasing without stationary points over the first subinterval, it is constant over the second one, and it is strictly monotone increasing without stationary points over the third subinterval. Each one or two of these parts may be empty or degenerate into a single point. The proof of this property is easy, when the function is differentiable. We consider functions, which are pseudoconvex with respect to the lower Dini directional derivative. This result follows from some known claims, but our theorem is a shot, directed to the target. We apply this characterization to obtain a complete characterization of strictly pseudoconvex functions. We also derive the respective results, when the function is radially lower semicontinuous in a real linear space. Several applications of the characterization are provided. A result due to Diewert, Avriel and Zang is extended to radially continuous functions.

In this paper, we introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is harmonized with the classical higher-order Fr\'echet directional derivative in the sense that both derivatives of the same order coincide if the last one exists. We obtain necessary and sufficient conditions of order $n$ ($n$ is a positive integer) for a local minimum and isolated local minimum of order $n$ of the given function in terms of these derivatives and subdifferentials. We do not require any restrictions on the function in our results. A notion of a higher-order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order $n$. Higher-order invex functions are defined. They are the largest class such that our necessary conditions for local minima are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order $n$., Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:1311.2367

In this paper, we provide some new necessary and sufficient conditions for pseudoconvexity and semistrict quasiconvexity of a given proper extended real-valued function in terms of the Clarke–Rockafellar subdifferential. Further, we extend to programs with pseudoconvex objective function two earlier characterizations of the solutions set of a set constrained nonlinear programming problem due to Mangasarian (Oper Res Lett 7:21–26, 1988). A positive function p appears in the most results. It is replaced by the number 1 if the function is convex and its domain of definition is convex, too.

A new notion of a second-order KT invex problem (P) with inequality constraints is introduced in this paper. This class of problems strictly includes the KT invex ones. Some properties of the second-order KT invex problems are presented. For example, (P) is second-order KT invex if and only if each point, which satisfies the second-order Kuhn-Tucker necessary optimality conditions, is a global minimizer. A problem with quasiconvex data is (second-order) KT invex if and only if it is (second-order) KT pseudoconvex.

In this paper we obtain second-order optimality conditions of Fritz John and Karush–Kuhn–Tucker types for the problem with inequality constraints in nonsmooth settings using a new second-order directional derivative of Hadamard type. We derive necessary and sufficient conditions for a point \({\bar x}\) to be a local minimizer and an isolated local one of order two. In the primal necessary conditions we suppose that all functions are locally Lipschitz, but in all other conditions the data are locally Lipschitz, regular in the sense of Clarke, Gâteaux differentiable at \({\bar x}\), and the constraint functions are second-order Hadamard differentiable at \({\bar x}\) in every direction. It is shown by an example that regularity and Gâteaux differentiability cannot be removed from the sufficient conditions.

In this article we introduce the notions of Kuhn-Tucker and Fritz John pseudoconvex nonlinear programming problems with inequality constraints. We derive several properties of these problems. We prove that the problem with quasiconvex data is (second-order) Kuhn-Tucker pseudoconvex if and only if every (second-order) Kuhn-Tucker stationary point is a global minimizer. We obtain respective results for Fritz John pseudoconvex problems. For the first-order case we consider Frechet differentiable functions and locally Lipschitz ones, for the second-order case Frechet and twice directionally differentiable functions.

In this paper we obtain second-order optimality conditions of Karush–Kuhn–Tucker type and Fritz John one for a problem with inequality constraints and a set constraint in nonsmooth settings using second-order directional derivatives. In the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable, but their gradients are not necessarily locally Lipschitz. In the sufficient conditions for a global minimum x ¯ we assume that the objective function is differentiable at x ¯ and second-order pseudoconvex at x ¯ , a notion introduced by the authors [I. Ginchev, V.I. Ivanov, Higher-order pseudoconvex functions, in: I.V. Konnov, D.T. Luc, A.M. Rubinov (Eds.), Generalized Convexity and Related Topics, in: Lecture Notes in Econom. and Math. Systems, vol. 583, Springer, 2007, pp. 247–264], the constraints are both differentiable and quasiconvex at x ¯ . In the sufficient conditions for an isolated local minimum of order two we suppose that the problem belongs to the class C 1 , 1 . We show that they do not hold for C 1 problems, which are not C 1 , 1 ones. At last a new notion parabolic local minimum is defined and it is applied to extend the sufficient conditions for an isolated local minimum from problems with C 1 , 1 data to problems with C 1 one.

Some necessary and sufficient optimality conditions for inequality constrained problems with continuously differentiable data were obtained in the papers [I. Ginchev and V.I. Ivanov, Second-order optimality conditions for problems with C$\sp{1}$ data, J. Math. Anal. Appl., v. 340, 2008, pp. 646--657], [V.I. Ivanov, Optimality conditions for an isolated minimum of order two in C$\sp{1}$ constrained optimization, J. Math. Anal. Appl., v. 356, 2009, pp. 30--41] and [V. I. Ivanov, Second- and first-order optimality conditions in vector optimization, Internat. J. Inform. Technol. Decis. Making, 2014, DOI: 10.1142/S0219622014500540]. In the present paper, we continue these investigations. We obtain some necessary optimality conditions of Karush--Kuhn--Tucker type for scalar and vector problems. A new second-order constraint qualification of Zangwill type is introduced. It is applied in the optimality conditions.

In this paper, we introduce a new second-order directional derivative and a second-order subdifferential of Hadamard type for an arbitrary nondifferentiable function. We derive second-order necessary and sufficient optimality conditions for a local and a global minimum and an isolated local minimum of second-order in unconstrained optimization. In particular, we obtain two results with strongly pseudoconvex functions. We also compare our conditions with the results of the recently published paper (Bednarik and Pastor, 2008) and a lot of other works, published in high level journals, and prove that they are particular cases of our necessary and sufficient ones. We prove that the necessary optimality conditions concern more functions than the conditions in terms of lower Dini directional derivative, even the optimality conditions with the last derivative can be applied to a function, which does not belong to some special class. At last, we apply our optimality criteria for scalar problems to derive necessary and sufficient optimality conditions in the cone-constrained vector optimization.

In this paper, we obtain higher order optimality conditions of Fritz John and Karush–Kuhn–Tucker types for the problem with inequality constraints in terms of higherorder Dini directional derivatives. In the necessary conditions, we suppose that the derivatives of order ( is a positive integer) and lower exist, and they are finite. In the sufficient ones, we assume additionally that the objective function is pseudoinvex of order with respect to some known map , and the constraints are prequasiinvex with respect to the same .

We derive necessary and sufficient conditions for optimality of a problem with a pseudoconvex objective function, provided that a finite number of solutions are known. In particular, we see that the gradient of the objective function at every minimizer is a product of some positive function and the gradient of the objective function at another fixed minimizer. We apply this condition to provide several complete characterizations of the solution sets of set-constrained and inequality-constrained nonlinear programming problems with pseudoconvex and second-order pseudoconvex objective functions in terms of a known solution. Additionally, we characterize the solution sets of the Stampacchia and Minty variational inequalities with a pseudomonotone-star map, provided that some solution is known.

Second-order optimality conditions for vector nonlinear programming problems with inequality constraints are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint qualification as a particular case. We obtain necessary and sufficient conditions for weak efficiency of problems with a second-order pseudoconvex vector objective function and quasiconvex constraints. We also derive Lagrange multiplier characterizations of the solution set of a scalar problem with a second-order pseudoconvex objective function and quasiconvex inequality constraints, provided that one of the solutions and the Lagrange multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we introduce a notion of a second-order KKT-pseudoconvex problem with inequality constraints. We derive sufficient and also necessary conditions for efficiency of second-order KKT-pseudoconvex problems. Three examples are presented., Comment: The submission contains 15 pages. I replaced the first version of the paper by another one, because I have published the introduced constraint qualification in another article. In this version it is replaced by a new second-order constraint qualification, and therefore a new theorem appears. I have added a new section with new results also

We introduce a notion of a second-order invex function. A Frechet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Frechet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type...

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.

In this paper we obtain first and second-order optimality conditions for an isolated minimum of order two for the problem with inequality constraints and a set constraint. First-order sufficient conditions are derived in terms of generalized convex functions. In the necessary conditions we suppose that the data are continuously differentiable. A notion of strongly KT invex inequality constrained problem is introduced. It is shown that each Kuhn–Tucker point is an isolated global minimizer of order two if and only if the problem is strongly KT invex. The article could be considered as a continuation of [I. Ginchev, V.I. Ivanov, Second-order optimality conditions for problems with C 1 data, J. Math. Anal. Appl. 340 (2008) 646–657].