Kuhn-Tucker Point Research Articles

Bilevel programming for the continuous transport network design problem

A Continuous Network Design Problem (CNDP) is to determine the set of link capacity expansions and the corresponding equilibrium flows for which the measures of performance index for the network is optimal. A bilevel programming technique can be used to formulate this equilibrium network design problem. At the upper level problem, the system performance index is defined as the sum of total travel times and investment costs of link capacity expansions. At the lower level problem, the user equilibrium flow is determined by Wardrop’s first principle and can be formulated as an equivalent minimization problem. In this paper we exploit a descent approach via the implementation of gradient-based methods to solve CNDP generally where the Karush–Kuhn–Tucker points can be obtained. Four variants of gradient-based methods are presented and numerical comparisons are widely made with the previous on three kinds of test networks. The proposed methods have achieved substantially better results in terms of the robustness to the initials and the computational efficiency in solving equilibrium assignment problems than did others especially when the congested road networks are considered.

Efficiency and generalised convexity in vector optimisation problems

AbstractThis paper gives a necessary and sufficient condition for a Kuhn-Tucker point of a non-smooth vector optimisation problem subject to inequality and equality constraints to be an efficient solution. The main tool we use is an alternative theorem which is quite different to a corresponding result by Xu.

Global Convergence of a Trust Region Algorithm for Nonlinear Inequality Constrained Optimization Problems

In the paper, a new trust region algorithm is given for nonlinear inequality constrained optimization problems. Motivated by a dual problem introduced by Han and Mangasarian [Han, S. P., Mangasarian, O. L. (1983). A dual differentiable exact penalty function. Math. Programming 25:293–306], which is a nonnegatively constrained maximization problem, we construct a trust region algorithm for solving the dual problem. At each iteration, we only need to minimize a quadratic subproblem with simple bound constraints. Under the condition that the iterate sequence generated by the algorithm is contained in some bounded closed set, any accumulation point of the sequence is a Karush–Kuhn–Tucker point of the original problem.

Convergence analysis of a nonlinear Lagrangian algorithm for nonlinear programming with inequality constraints

In this paper, we establish a nonlinear Lagrangian algorithm for nonlinear programming problems with inequality constraints. Under some assumptions, it is proved that the sequence of points, generated by solving an unconstrained programming, convergents locally to a Kuhn-Tucker point of the primal nonlinear programming problem.

On the Complexity of Equalizing Inequalities

We study two approaches to replace a finite mathematical programming problem with inequality constraints by a problem that contains only equality constraints. The first approach lifts the feasible set into a high-dimensional space by the introduction of quadratic slack variables. We show that then not only the number of critical points but also the topological complexity of the feasible set grow exponentially. On the other hand, the second approach bases on an interior point technique and lifts an approximation of the feasible set into a space with only one additional dimension. Here only Karush–Kuhn–Tucker points with respect to the positive and negative objective function in the original problem give rise to critical points of the smoothed problem, so that the number of critical points as well as the topological complexity can at most double.

On local search for the generalized graph coloring problem

Given an edge-weighted graph and an integer k, the generalized graph coloring problem is the problem of partitioning the vertex set into k subsets so as to minimize the total weight of the edges that are included in a single subset. We recall a result on the equivalence between Karush–Kuhn–Tucker points for a quadratic programming formulation and local optima for the simple flip-neighborhood. We also show that the quality of local optima with respect to a large class of neighborhoods may be arbitrarily bad and that some local optima may be hard to find.

The Pre-Variational Problems and the Constrained Mathematical Programming Problem

In this paper relationships between the solutions of the Variational- Like and Pre-Variational Problems with those of the Constrained Mathematical Programming Problems have been established. Sufficient conditions are obtained for Kuhn-Tucker points to be minima of the Constrained Mathematical Programming Problem, through the solutions of the Pre-Variational Problems.

Inequality-Equality Constrained Optimization: An Analytical Robustness Comparison of a Feasibility Method Versus L1 Sequential Quadratic Programming

In this work we present an analytical robustness comparison of two methods for inequality-equality constrained optimization or nonlinear programming. The methods compared are (1) a Feasibility Method (FM) and (2) rudimentary sequential quadratic programming with an L1 merit function (L1-SQP). We then also make note of a very recent global convergence result (similar to that of FM) for a new Filter-type SQP algorithm. And, we claim no analytical robustness advantage of FM over the Filter-type SQP algorithm. The problem statement assumptions include nonstationarity of constraint error norms except at zero constraint error, without which we are not aware of any algorithm that is provably guaranteed to converge to a tolerance-feasible stationary point of a penalty function or a Kuhn-Tucker point. Global convergence of FM is proved analytically. Rudimentary L1-SQP is shown to exhibit potential failure even from a feasible starting point, due to an onset of infeasible subproblems.

Bilevel programming applied to optimising urban transportation

This paper outlines a multi-modal, elastic, equilibrium transportation model in which signal green-times and prices charged to traverse a route (public transport fares, parking charges or road-use charges) are explicitly included. An algorithm is specified which, for a fairly general objective function, continually moves current traffic flows, green-times and prices within the model toward locally optimal values while taking account of users’ responses. The directions of movement of current traffic flows, green-times and prices are determined by solving linear approximations to the actual problem. The results of applying a simplified form of the algorithm to a small network model with five routes and two signal-controlled junctions are given. It is proved that under realistic conditions the sequence of (traffic flows, green-times, prices) triples generated by the algorithm does indeed approach those triples which possess a reasonable local optimality property. However the optimal control problem discussed here is non-convex and just a Karush–Kuhn–Tucker point is the “answer” sought.

The convergence of a dual algorithm for nonlinear programming

A dual algorithm based on the smooth function proposed by Polyak (1988) is constructed for solving nonlinear programming problems with inequality constraints. It generates a sequence of points converging locally to a Kuhn-Tucker point by solving an unconstrained minimizer of a smooth potential function with a parameter. We study the relationship between eigenvalues of the Hessian of this smooth potential function and the parameter, which is useful for analyzing the effectiveness of the dual algorithm.

Optimization of Area Traffic Control for Equilibrium Network Flows

A bilevel programming approach is used to tackle an optimization problem for area traffic control and equilibrium flows. The signal timing plan is defined by common cycle time, and by starts and durations of greens. The system performance index is defined as the sum of a weighted linear combination of rate of delay and number of stops per unit time for all traffic streams, which is evaluated by the traffic model from TRANSYT. User equilibrium traffic assignment is formulated as a variational inequality problem. Approximate mathematical expressions for various components of the performance index and the average delay to a vehicle at the downstream junction in the TRANSYT model for both undersaturated and oversaturated links have been derived. For a locally optimal search, the gradient projection method is used in deciding whether feasible descent directions leading to a Karush–Kuhn–Tucker point, which is potentially a local optimum, can be identified. A global search heuristic is proposed in this paper by which successively better Karush–Kuhn–Tucker points can be found with reasonable computation effort. The mixed search procedure, including the locally optimal search and global search heuristic, is proposed. Encouraging results for Allsop and Charlesworth's network have confirmed that the approximately optimizing mixed search procedure in solving the area traffic control optimization problem can achieve substantially better results than does the nonoptimizing calculations of mutually consistent signal timings and link flows.

Invex Functions and Generalized Convexity in Multiobjective Programming

Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that the equivalence between minima and stationary points or Kuhn–Tucker points (depending on the case) remains true if we optimize several objective functions instead of one objective function. To this end, we define accurately stationary points and Kuhn–Tucker optimality conditions for multiobjective programming problems. We see that the Martin results cannot be improved in mathematical programming, because the new types of generalized convexity that have appeared over the last few years do not yield any new optimality conditions for mathematical programming problems.

Erratum: An SQP Algorithm for Finely Discretized Continuous Minimax Problems and Other Minimax Problems with Many Objective Functions

Previous article Full AccessErratum: An SQP Algorithm for Finely Discretized Continuous Minimax Problems and Other Minimax Problems with Many Objective FunctionsJian L. Zhou and André L. TitsJian L. Zhou and André L. Titshttps://doi.org/10.1137/S1052623496313659PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractAn error is pointed out in the local convergence proof in the quoted paper [J. L. Zhou and A. L. Tits, SIAM J. Optim., 6 (1996), pp. 461--487]. A correct proof is given.[1] S. M. Robinson, Perturbed Kuhn–Tucker points and rates of convergence for a class of nonlinear‐programming algorithms, Math. Programming, 7 (1974), pp. 1–16. mhp MHPGA4 0025-5610 Math. Program. CrossrefGoogle Scholar[2] Jian Zhou and , André Tits, An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions, SIAM J. Optim., 6 (1996), 461–487 97a:90101 LinkISIGoogle ScholarKeywordscontinuous minimaxsemi-infinite programmingmany constraintssequential quadratic programmingdiscretizationglobal convergence Previous article FiguresRelatedReferencesCited byDetails A box moments approach for the time-variant hybrid reliability assessment14 September 2021 | Structural and Multidisciplinary Optimization, Vol. 64, No. 6 Cross Ref SIPAMPLACM Transactions on Mathematical Software, Vol. 30, No. 1 Cross Ref Design optimization of induction motors for aerospace applications Cross Ref Volume 8, Issue 1| 1998SIAM Journal on Optimization History Published online:31 July 2006 InformationCopyright © 1998 Society for Industrial and Applied MathematicsKeywordscontinuous minimaxsemi-infinite programmingmany constraintssequential quadratic programmingdiscretizationglobal convergenceMSC codes49M0749M3749M3965K0590C0690C3090C34PDF Download Article & Publication DataArticle DOI:10.1137/S1052623496313659Article page range:pp. 284-285ISSN (print):1052-6234ISSN (online):1095-7189Publisher:Society for Industrial and Applied Mathematics

Constrained Optimization via Stochastic Approximation with a Simultaneous Perturbation Gradient Approximation

The paper deals with a projection algorithm for stochastic approximation using simultaneous perturbation gradient approximation for optimization under inequality constraints where no direct gradient of the loss function is available and the inequality constraints are given as explicit functions of the optimization parameters. It is shown that under application of the projection algorithm, the parameter iterate converges almost surely to a Kuhn-Tucker point. The procedure is illustrated by a numerical example.

Constrained optimization via stochastic approximation with a simultaneous perturbation gradient approximation

This paper deals with a projection algorithm for stochastic approximation using simultaneous perturbation gradient approximation for optimization under inequality constraints where no direct gradient of the loss function is available and the inequality constraints are given as explicit functions of the optimization parameters. It is shown that, under application of the projection algorithm, the parameter iterate converges almost surely to a Kuhn-Tucker point. The procedure is illustrated by a numerical example.

A Modified SQP Method and Its Global Convergence

The sequential quadratic programming method developed by Wilson, Han and Powell may fail if the quadratic programming subproblems become infeasible or if the associated sequence of search directions is unbounded. In [1], Han and Burke give a modification to this method wherein the QP subproblem is altered in a way which guarantees that the associated constraint region is nonempty and for which a robust convergence theory is established. In this paper, we give a modification to the QP subproblem and provide a modified SQP method. Under some conditions, we prove that the algorithm either terminates at a Kuhn–Tucker point within finite steps or generates an infinite sequence whose every cluster is a Kuhn–Tucker point. Finally, we give some numerical examples.

A combined homotopy interior point method for general nonlinear programming problems

A combined homotopy interior point method for solving general nonlinear programming is proposed. The algorithm generated by this method to Kuhn-Tucker points of the general nonlinear programming problems is proved to be globally convergent, under the “normal cone condition” about the constraints, probably without the convexity.

Second-order sufficient optimality conditions for local and global nonlinear programming

This paper presents a new approach to the sufficient conditions of nonlinear programming. Main result is a sufficient condition for the global optimality of a Kuhn-Tucker point. This condition can be verified constructively, using a novel convexity test based on interval analysis, and is guaranteed to prove global optimality of strong local minimizers for sufficiently narrow bounds. Hence it is expected to be a useful tool within branch and bound algorithms for global optimization.

Nonconvex process optimization

Difficulties associated with nonconvexity in successive quadratic programming (SQP) methods are studied. It is shown that projected indefiniteness of the Hessian matrix of the Lagrangian function can (i) place restrictions on the order in which inequalities can be added or deleted from the active set, (ii) generate redundant active sets whose resolution is nontrivial, (iii) give rise to quadratic programming (QP) subproblems that have multiple Kuhn-Tucker points, and (iv) produce nondescent directions in the SQP method that can lead to failure. Related issues concerned with the use of feasible or infeasible starting points for the iterative quadratic programs, forcing positive definiteness to ensure convexity, and using iterative methods to solve the linear Kuhn-Tucker conditions associated with the QP subproblems are also studied. A new active set strategy that (i) monitors projected indefiniteness to guide the addition of constraints to the active set, (ii) permits line searching for negative values of the line search parameter, and (iii) does not necessarily delete active constraints with incorrect Kuhn-Tucker multipliers is proposed. Constraint redundancy is circumvented using an algorithm that identifies all nontrivial redundant subsets of smallest size and determines which, if any, exchanges are justified. Nondescent in the NLP's is resolved using a linear programming (LP)-based trust region method that guarantees descent regardless of merit function. It is also shown that there is no justification for using feasible starting points at the QP level of the calculations, that forcing positive definiteness to ensure convexity can cause termination at undesired solutions, and that the use of iterative methods to solve the linear Kuhn-Tucker equations for the QP's can cause a deterioration in numerical performance. Many small chemical process examples are used to highlight difficulties so that geometric illustrations can be used while heat exchange network design and distillation operations examples are used to show that these same difficulties carry over to larger problems.

Global Optimality Conditions and their Geometric Interpretation for the Chemical and Phase Equilibrium Problem

A general class of nonlinear optimization problems motivated by the chemical and phase equilibrium problem in chemical thermodynamics is discussed. The relationships between Kuhn–Tucker points and the global minimum are investigated. The relationships are interpreted in terms of common tangent planes to a function with domain in $\mathcal{R}^{N - 1} $ associated with the objective function, whose domain is in $\mathcal{R}^N $. Necessary and sufficient conditions for a global minimum are established, which we call the reaction tangent-plane criterion. The conditions related to the common tangent planes may be considered separately from the feasibility conditions, which allows a novel geometric interpretation of the overall optimality conditions. Illustrative examples are provided of systems involving up to three chemical species.