Saddle Point Condition Research Articles

Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming

This paper aims at showing that the class of augmented Lagrangian functions for nonlinear semidefinite programming problems can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem. By means of the image space analysis, a global saddle point condition for the augmented Lagrangian function is investigated. It is shown that the existence of a saddle point is equivalent to a regular nonlinear separation of two suitable subsets of the image space. Without requiring the strict complementarity, it is proved that, under second order sufficiency conditions, the augmented Lagrangian function admits a local saddle point. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set. Motivated by the result on global saddle points, we propose two modified primal-dual methods based on the augmented Lagrangian using different strategies and prove their convergence to a global solution and the optimal value of the original problem without requiring the boundedness condition of the multiplier sequence.

Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis

The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.

A Disaggregate Residential Equilibrium Assignment Model

This paper presents a set of hypotheses and a mathematical formulation in order to model the equilibrium between supply and demand on an urban housing market. Housing supply is assumed to be price-elastic, in a way that can vary for each location and each dwelling type. Household housing demand is disaggregated on the basis of employment area, category of activity, income and household size. The market equilibrium is characterized by a set of primal-dual conditions, then by a variational inequality whose continuity guarantees the existence of an equilibrium. In the monocentric case, the primal-dual conditions are the saddle point conditions of a constrained maximization program the concavity of which implies the uniqueness of the equilibrium. We propose a resolution algorithm that is derived from traffic assignment on a transportation network. Last, we explore the outcomes of the model in a demonstration application and analyze the sensitivity of the equilibrium to our supply and demand hypotheses.

Nonlinear Separation Approach to Constrained Extremum Problems

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.

Special Saddle Point Conditions for Max-Min Discrete Control Problem and Algorithms for Determining the Optimal Strategies

In this contribution, a special concept of antagonistic games to finite state space discrete control problems with an integral-time cost optimization criterion on the set of trajectories is applied. Based on such a specific approach a new class of dynamic positional games is formulated and studied. Saddle point conditions for the considered class of games are derived and algorithms for determining the optimal strategies of the players are proposed.

Nonlinear separation in the image space with applications to penalty methods

In this article we analyse a nonlinear separation scheme in the image space associated with an infinite-dimensional cone constrained extremum problem, and, in particular, we consider the applications to nonlinear Lagrangian duality and exact and inexact exterior penalty methods. Duality and exact penalty methods arise from the existence of a regular nonlinear separation, which is shown to be equivalent to a saddle point condition for a generalized Lagrangian function associated with the given problem, while inexact penalty methods are closely related with the existence of an asymptotic nonlinear separation.

A new 2d/4d duality via integrability

We prove a duality, recently conjectured in arXiv:1103.5726, which relates the F-terms of supersymmetric gauge theories defined in two and four dimensions respectively. The proof proceeds by a saddle point analysis of the four-dimensional partition function in the Nekrasov-Shatashvili limit. At special quantized values of the Coulomb branch moduli, the saddle point condition becomes the Bethe Ansatz Equation of the SL(2) Heisenberg spin chain which coincides with the F-term equation of the dual two-dimensional theory. The on-shell values of the superpotential in the two theories are shown to coincide in corresponding vacua. We also identify two-dimensional duals for a large set of quiver gauge theories in four dimensions and generalize our proof to these cases.

Saddle points of general augmented Lagrangians for constrained nonconvex optimization

Local and global saddle point conditions for a general augmented Lagrangian function proposed by Mangasarian are investigated in the paper for inequality and equality constrained nonconvex optimization problems. Under second order sufficiency conditions, it is proved that the augmented Lagrangian admits a local saddle point, but without requiring the strict complementarity condition. The existence of a global saddle point is then obtained under additional assumptions that do not require the compactness of the feasible set and the uniqueness of global solution of the original problem.

On zero duality gap in nonconvex quadratic programming problems

We present in this paper new sufficient conditions for verifying zero duality gap in nonconvex quadratically/linearly constrained quadratic programs (QP). Based on saddle point condition and conic duality theorem, we first derive a sufficient condition for the zero duality gap between a quadratically constrained QP and its Lagrangian dual or SDP relaxation. We then use a distance measure to characterize the duality gap for nonconvex QP with linear constraints. We show that this distance can be computed via cell enumeration technique in discrete geometry. Finally, we revisit two sufficient optimality conditions in the literature for two classes of nonconvex QPs and show that these conditions actually imply zero duality gap.

Saddlepoint condition on a predictor to reconfirm the need for the assumption of a prior distribution

Saddlepoint conditions on a predictor are introduced and developed to reconfirm the need for the assumption of a prior distribution in constructing a useful inferential procedure. A condition yields that the predictor induced from the maximum likelihood estimator is the worst under a loss, while the predictor induced from a suitable posterior mean is the best. This result indicates the promising role of Bayesian criteria, such as the deviance information criterion (DIC). As an implication, we critique the conventional empirical Bayes method because of its partial assumption of a prior distribution.

Image Space Analysis for Vector Variational Inequalities with Matrix Inequality Constraints and Applications

In this paper, vector variational inequalities (VVI) with matrix inequality constraints are investigated by using the image space analysis. Linear separation for VVI with matrix inequality constraints is characterized by using the saddle-point conditions of the Lagrangian function. Lagrangian-type necessary and sufficient optimality conditions for VVI with matrix inequality constraints are derived by utilizing the separation theorem. Gap functions for VVI with matrix inequality constraints and weak sharp minimum property for the solutions set of VVI with matrix inequality constraints are also considered. The results obtained above are applied to investigate the Lagrangian-type necessary and sufficient optimality conditions for vector linear semidefinite programming problems as well as VVI with convex quadratic inequality constraints.

Minimizing measures of risk by saddle point conditions

The minimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions. This new problem involves Banach spaces but a mean value theorem for risk measures is stated, and this simplifies the dual problem. Then, optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable. This characterization is significantly different if one compares it with previous literature. Furthermore, the saddle point condition very easily applies in practice. Four applications in finance and insurance are presented.

Separation Approach for Augmented Lagrangians in Constrained Nonconvex Optimization

This paper aims at showing that the class of augmented Lagrangian functions, introduced by Rockafellar and Wets, can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem; hence, it is part of a more general theory. By means of the image space analysis, local and global saddle-point conditions for the augmented Lagrangian function are investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Under second-order sufficiency conditions in the image space, it is proved that the augmented Lagrangian admits a local saddle point. The existence of a global saddle point is then obtained under additional assumptions that do not require the compactness of the feasible set.

Effective method for constrained minimum — reverse bridge theorem

We propose reverse bridge theorem (RBTH) and give the demonstration in this work. RBTH is an effective method for constrained minimum. It is necessary and sufficient condition. RBTH bases on Extended saddle-point condition (ESPC), which is proposed by Yixin Chen. SGPlan according to ESPC dose the best in satisfying the International Planning Competition 2006 and in Suboptimal Metric Track of International Planning Competition 2004. RBTH comes into all advantage of ESPC and is different from ESPC, which can be concluded from two points. First, the core inequality of ESPC is formed by two sub-inequalities, and the core inequality of RBTH contains only one sub-inequality and one sub-equality. Generally, equality is easier to handle than inequality. RBTH should be better for planner than ESPC. The core inequality changes and is no longer saddle shape. We analyze why the core inequality changes. Second, continuous RBTH (CRBTH) does not need constraint-qualification condition which is necessary for continuous ESPC. This can be authenticated in two respects. One is the proof of CRBTH; the other is KKT which has relationship with ESPC. RBTH is real necessary and sufficient condition and ESPC is not. So RBTH can be used more widely. Besides, we show some errors in ESPC. We define point domain and Shrink-point domain to explain error. We also analyze the cause of errors, and provide an example.

$B$-SEMIPREINVEX FUNCTIONS AND VECTOR OPTIMIZATION PROBLEMS IN BANACH SPACES

In this paper, we extend the scalar-valued $B$-semipreinvex functions and vector-valued preinvex functions to the cases of vector-valued $B$-semipreinvex functions in Banach spaces. We investigate some properties for the vector-valued $B$-semipreinvex functions and consider a new class of vector-valued nonsmooth programming problems in which functions are locally Lipschitz. In terms of the Ralph vector sub-gradient, we obtain the generalized Kuhn-Tucker type sufficient optimality conditions and saddle point condition. Also, a generalized Mond-Weir type dual is formulated and some duality theorems are established involving locally Lipschitz $B$-semipreinvex functions for the pair of primal and dual programming. The results presented in this paper generalize some main results of Kuang and Batista Dos Santoset, Osuna-Gomez, Rojas-Medar and Rufian-Lizana.

Quantum statistical mechanics of gases in terms of dynamical filling fractions and scattering amplitudes

We develop a finite temperature field theory formalism in any dimension that has the filling fractions as the basic dynamical variables. The formalism efficiently decouples zero temperature dynamics from the quantum statistical sums. The zero temperature ‘data’ are the scattering amplitudes. A saddle-point condition leads to an integral equation which is similar in spirit to the thermodynamic Bethe ansatz for integrable models, and effectively resums infinite classes of diagrams. We present both relativistic and non-relativistic versions.

Pairing correlations and thermodynamical quantities inMo96,97

The nuclear level densities of 96,97 Mo are calculated in the framework of superconducting theory. The parameters of nuclear level density are so chosen that the saddle point conditions are satisfied and the best fit to the experimental data yields. Then, using these parameters the energy, the entropy and the spin cut-off factor are calculated as a function of temperature. The curves show structures, reflecting the phase transition from a correlated to an uncorrelated phase. The critical temperature for quenching of pairing correlations is found at T c ∼ 0.7-0.9 MeV.

MSSM flat direction inflation: slow roll, stability, fine-tuning and reheating

We consider low scale slow roll inflation driven by the gauge-invariant flatdirections udd and LLe of the minimally supersymmetric standard model at thevicinity of a saddle point of the scalar potential. We study the stability of thesaddle point and the slow roll regime by considering radiative and supergravitycorrections. The latter are found to be harmless, but the former require a modestfine-tuning of the saddle point condition. We show that, while the inflaton decaysalmost instantly, full thermalization occurs late, typically at a temperatureT≈107 GeV, so that there is no gravitino problem. We also compute the renormalization grouprunning of the inflaton mass and relate it to slepton masses that may be within thereach of the LHC and could be precisely determined in a future linear colliderexperiment.

Hedging Interest Rate Risk by Optimization in Banach Spaces

This paper addresses the hedging of bond portfolios interest rate risk by drawing on the classical one-period no-arbitrage approach of financial economics. Under quite weak assumptions, several maximin portfolios are introduced by means of semi-infinite mathematical programming problems. These problems involve several Banach spaces; consequently, infinite-dimensional versions of classical algorithms are required. Furthermore, the corresponding solutions satisfy a saddle-point condition illustrating how they may provide appropriate hedging with respect to the interest rate risk.

B-SEMIPREINVEX FUNCTIONS AND VECTOR OPTIMIZATION PROBLEMS IN BANACH SPACES

In this paper, we extend the scalar-valued $B$-semipreinvex functions and vector-valued preinvex functions to the cases of vector-valued $B$-semipreinvex functions in Banach spaces. We investigate some properties for the vector-valued $B$-semipreinvex functions and consider a new class of vector-valued nonsmooth programming problems in which functions are locally Lipschitz. In terms of the Ralph vector sub-gradient, we obtain the generalized Kuhn-Tucker type sufficient optimality conditions and saddle point condition. Also, a generalized Mond-Weir type dual is formulated and some duality theorems are established involving locally Lipschitz $B$-semipreinvex functions for the pair of primal and dual programming. The results presented in this paper generalize some main results of Kuang and Batista Dos Santoset, Osuna-Gomez, Rojas-Medar and Rufian-Lizana.