Metric aspects of conic surfaces

We give a survey on various results regarding the metric aspects of conic surfaces with emphasis on the prescribing curvature problem for conic surfaces.     

Stability of a superposition of shock waves with contact discontinuities for systems of viscous conservation laws

In this paper, we show the large time asymptotic nonlinear stability of a superposition of viscous shock waves with viscous contact waves for systems of viscous conservation laws with small initial perturbations, provided that the strengths of these viscous waves are small with the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a new estimate on the heat equation.     

Zeta regularized determinants for conic manifolds

We study (relative) zeta regularized determinants of Laplace type operators on compact conic manifolds. We establish gluing formulae for relative zeta regularized determinants. For arbitrary self-adjoint extensions of the Laplace-Beltrami operator, we express the relative ζ-determinants for these as a ratio of the determinants of certain finite matrices. For the self-adjoint extensions corresponding to Dirichlet and Neumann conditions, the formula is particularly simple and elegant.     

Minimal conic quadratic reformulations and an optimization model

In this paper, we consider a particular form of inequalities which involves product of multiple variables with rational exponents. These inequalities can equivalently be represented by a number of conic quadratic forms called cone constraints. We propose an integer programming model and a heuristic algorithm to obtain the minimum number of cone constraints which equivalently represent the original inequality. The performance of the proposed algorithm and the computational effect of reformulations are numerically illustrated.     

Theorems of the alternative for conic integer programming

Farkas’ Lemma is a foundational result in linear programming, with implications in duality, optimality conditions, and stochastic and bilevel programming. Its generalizations are known as theorems of the alternative. There exist theorems of the alternative for integer programming and conic programming. We present theorems of the alternative for conic integer programming. We provide a nested procedure to construct a function that characterizes feasibility over right-hand sides and can determine which statement in a theorem of the alternative holds.     

Existence of multidimensional shock waves and phase boundaries

In this paper, a kind of Riemann problem for the Euler equations in a van der Waals fluid is considered. We constructed the weak solution in multidimensional space which contains one shock front and one subsonic phase boundary. We mainly follow the arguments of Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95; A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1-93] and Métivier's [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479] work. The linear stability results are based on Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95] work for the single shock front and Wang and Xin's [Y.-G. Wang, Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sin. 19 (2003) 529-558] work for the single phase boundary. The initial boundary value problem concerned in this paper is different from the boundary value problem for double shock fronts concerned in [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479], we slightly modified Métivier's frame work to establish the existence for the solution to the nonlinear problem.     

Inverse conic programming with applications

Linear programming duality yields efficient algorithms for solving inverse linear programs. We show that special classes of conic programs admit a similar duality and, as a consequence, establish that the corresponding inverse programs are efficiently solvable. We discuss applications of inverse conic programming in portfolio optimization and utility function identification.     

The propagation of impact-induced tensile waves in a kind of phase-transforming materials

This paper concerns the propagation of impact-generated tensile waves in a one-dimensional bar made of a kind of phase-transforming materials, for which the stress–strain curve changes from concave to convex as the strain increases. We use the fully nonlinear curve instead of approximating it by a tri-linear curve as often used in literature. The governing system of partial differential equations is quasi-linear and hyperbolic–elliptic. It is well known that the standard form of the initial-boundary value problem corresponding to impact is not well-posed at all levels of loading. In this paper, we describe in detail the propagation of impact-induced tensile waves for all levels. In particular, by means of the uniqueness condition on phase boundary derived recently, we construct a physical solution of the initial-boundary value problem mentioned above, and analyze the geometrical structure and behavior of the physical solution.     

Universal duality in conic convex optimization

Given a primal-dual pair of linear programs, it is well known that if their optimal values are viewed as lying on the extended real line, then the duality gap is zero, unless both problems are infeasible, in which case the optimal values are +∞ and −∞. In contrast, for optimization problems over nonpolyhedral convex cones, a nonzero duality gap can exist when either the primal or the dual is feasible. For a pair of dual conic convex programs, we provide simple conditions on the ``constraint matrices' and cone under which the duality gap is zero for every choice of linear objective function and constraint right-hand side. We refer to this property as ``universal duality'. Our conditions possess the following properties: (i) they are necessary and sufficient, in the sense that if (and only if) they do not hold, the duality gap is nonzero for some linear objective function and constraint right-hand side; (ii) they are metrically and topologically generic; and (iii) they can be verified by solving a single conic convex program. We relate to universal duality the fact that the feasible sets of a primal convex program and its dual cannot both be bounded, unless they are both empty. Finally we illustrate our theory on a class of semidefinite programs that appear in control theory applications. This work was supported by a fellowship at the University of Maryland, in addition to NSF grants DEMO-9813057, DMI0422931, CUR0204084, and DoE grant DEFG0204ER25655. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation or those of the US Department of Energy.     

A nonmonotone trust-region method of conic model for unconstrained optimization

In this paper, we present a nonmonotone trust-region method of conic model for unconstrained optimization. The new method combines a new trust-region subproblem of conic model proposed in [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231] with a nonmonotone technique for solving unconstrained optimization. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the method of [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231].     

New stopping criteria for detecting infeasibility in conic optimization

Detecting infeasibility in conic optimization and providing certificates for infeasibility pose a bigger challenge than in the linear case due to the lack of strong duality. In this paper we generalize the approximate Farkas lemma of Todd and Ye (Math Program 81:1–22, 1998) from the linear to the general conic setting, and use it to propose stopping criteria for interior point algorithms using self-dual embedding. The new criteria can identify if the solutions have large norm, thus they give an indication of infeasibility. The modified algorithms enjoy the same complexity bounds as the original ones, without assuming that the problem is feasible. Issues about the practical application of the criteria are also discussed. The authors were supported by the Canada Research Chairs program, NSERC Discovery Grant #5-48923 and MITACS.     

A copositive Farkas lemma and minimally exact conic relaxations for robust quadratic optimization with binary and quadratic constraints

We present a new copositive Farkas lemma for a general conic quadratic system with binary constraints under a convexifiability requirement. By employing this Farkas lemma, we establish that a minimally exact conic programming relaxation holds for a convexifiable robust quadratic optimization problem with binary and quadratic constraints under a commonly used ellipsoidal uncertainty set of robust optimization. We then derive a minimally exact copositive relaxation for a robust binary quadratic program with conic linear constraints where the convexifiability easily holds.     

The existence and stability of quasi-steady periodic voidage waves in a fluidized bed

Using the Hopf bifurcation theorem we are able to show the existence of a one-parameter family of quasi-steady periodic solutions to the nonlinear equations of motion governing the one-dimensional flow in a fluidized bed at flow rates for which the uniformly fluidized state is unstable. We are then able to obtain uniformly valid expansions for these periodic solutions close to the bifurcation point using the method of multiple scales, and extend these results by numerical integrations for certain values of the parameters. The bifurcation theorem also identifies a flow rateE ^*such that for flow rates less thanE ^*the bifurcation is supercritical while for flow rates greater thanE ^*it is subcritical.Having established that periodic solutions can exist, we then discuss the temporal stability of these periodic voidage waves by considering the weakly nonlinear evolution of a slowly varying wave train at flow rates close to critical stability. For flow rates close to the lower critical stability point we find that the periodic solutions are stable only for flow rates such that the uniform state is unstable and then provided their wavelength <^*. For flow rates close to the upper critical stability point we find, as well as there being a wavelength such that the periodic solutions are stable only if , that there is a further parameter _u with the condition _u 1 needed for stability.

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Résumé En utilisant le théorème de bifurcation de Hopf nous démontrons l'existence d'une famille à un paramètre de solutions périodiques quasi-stationnaires aux équations non-linéaires qui gouvernent l'écoulement uni-dimensionnel dans un lit fluidisé à des vitesses pour lesquelles l'état uniformément fluidisé est instable. En utilisant la méthode des échelles multiples, nous présentons des expansions uniformément valables pour ces solutions périodiques au voisinage du point de bifurcation, et l'extension de ces résultats par des intégrations numériques pour quelques valeurs de paramètres. Le théorème de bifurcation permet d'identifier une vitesseE ^*telle que, pour des vitesses inférieures àE ^* la bifurcation est surcritique, tandis que pour des vitesses supérieures àE ^* elle est souscritique.Une fois établi que des solutions périodiques peuvent exister, nous sommes à même de discuter la stabilité temporelle de ces ondes de vide périodiques en considérant l'évolution faiblement non-linéaire d'un train d'ondes avec variation lente près de la stabilité critique. Pour des vitesses proches du point inférieur de stabilité critique, nous établissons que les solutions périodiques sont stables seulement à des vitesses telles que l'état uniforme est instable, et à condition que leur longueur d'onde <^*. Pour des vitesses proches du point supérieur de stabilité critique, nous découvrons que, en plus d'une longueur d'onde telle que les solutions périodiques sont stables seulement si , il existe un autre paramètre _u avec la condition _u 1 nécessaire pour qu'il y ait stabilité.

     

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

We describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation.     

Adjustable robust counterpart of conic quadratic problems

This paper presents an approximate affinely adjustable robust counterpart for conic quadratic constraints. The theory is applied to obtain robust solutions to the problems of subway route design with implementation errors and a supply chain management with uncertain demands. Comparison of the adjustable solutions with the nominal and non-adjustable robust solutions shows that the adjustable (dynamic) robust solution maintains feasibility for all possible realizations, while being less conservative than the usual (static) robust counterpart solution.     

The stability of quasi-transverse shock waves in anisotropic elastic media

The stability of weak quasi-transverrse shock waves in a weakly anistropic elastic medium with respect to arbitrarily oriented perturbations is investigated in the linear approximation. It is shown that fast quasi-transverse shock waves are stable.     

Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly feasible points in both the primal and dual spaces can be defined. A second benefit of the modification is an improvement in the complexity analysis of conic cutting surface algorithms. Complexity results for conic cutting surface algorithms proved to date have depended on a condition number of the added constraints. The proposed modification of the constraints leads to a stronger result, with the convergence of the resulting algorithm not dependent on the condition number. Research supported in part by NSF grant number DMS-0317323. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.