Linear Independence Constraint Qualification Research Articles

An Augmented Lagrangian based Semismooth Newton Method for a Class of Bilinear Programming Problems

This paper proposes a semismooth Newton method for a class of bilinear programming problems (BLPs) based on the augmented Lagrangian, in which the BLPs are reformulated as a system of nonlinear equations with original variables and Lagrange multipliers. Without strict complementarity, the convergence of the method is studied by means of theories of semismooth analysis under the linear independence constraint qualification and strong second order sufficient condition. At last, numerical results are reported to show the performance of the proposed method.

On the linear independence constraint qualification in disjunctive programming

Mathematical programmes with disjunctive constraints (MPDCs for short) cover several different problem classes from nonlinear optimization including complementarity-, vanishing-, cardinality- and switching-constrained optimization problems. In this paper, we introduce an abstract but reasonable version of the prominent linear independence constraint qualification which applies to MPDCs. Afterwards, we derive first- and second-order optimality conditions for MPDCs under validity of this constraint qualification based on so-called strongly stationary points. Finally, we apply our findings to some popular classes of disjunctive programmes and compare the obtained results to those ones available in the literature. Particularly, new second-order optimality conditions for mathematical programmes with switching constraints are by-products of our approach.

The rate of convergence of proximal method of multipliers for nonlinear semidefinite programming

ABSTRACTThe proximal method of multipliers was proposed by Rockafellar [Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math Oper Res. 1976;1:97–116] for solving convex programming and it is a kind of proximal point method applied to convex programming. In this paper, we apply this method for solving nonlinear semidefinite programming problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under the linear independence constraint qualification and the strong second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for a nonlinear semidefinite programming problem, is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold . Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to .

The rate of convergence of proximal method of multipliers for equality constrained optimization problems

The proximal method of multipliers was proposed by Rockafellar (Math Oper Res 1:97–116, 1976) for solving convex programming and it is a kind of proximal point method for convex programming. In this paper, we apply this method for equality constrained optimization problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under linear independence constraint qualification and the second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for the equality constrained optimization problem, is linear and the ratio constant is proportional to 1 / c, where c is the penalty parameter that exceeds a threshold $$c_*> 0$$ . Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to $$+\infty $$ .

On the cost of solving augmented Lagrangian subproblems

At each iteration of the augmented Lagrangian algorithm, a nonlinear subproblem is being solved. The number of inner iterations (of some/any method) needed to obtain a solution of the subproblem, or even a suitable approximate stationary point, is in principle unknown. In this paper we show that to compute an approximate stationary point sufficient to guarantee local superlinear convergence of the augmented Lagrangian iterations, it is enough to solve two quadratic programming problems (or two linear systems in the equality-constrained case). In other words, two inner Newtonian iterations are sufficient. To the best of our knowledge, such results are not available even under the strongest assumptions (of second-order sufficiency, strict complementarity, and the linear independence constraint qualification). Our analysis is performed under second-order sufficiency only, which is the weakest assumption for obtaining local convergence and rate of convergence of outer iterations of the augmented Lagrangian algorithm. The structure of the quadratic problems in question is related to the stabilized sequential quadratic programming and to second-order corrections.

Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method

The existence of complementarity constraints causes the difficulties for studying mathematical programs with second-order cone complementarity constraints, since the standard constraint qualification, such as Robinson’s constraint qualification, is invalid. Therefore, various stationary conditions including strong, Mordukhovich and Clarke stationary conditions have been proposed, according to different reformulations of the second-order cone complementarity constraints. In this paper, we present a new reformulation of this problem by taking into consideration the Jordan algebra associated with the second-order cone. It ensures that the classical Karush–Kuhn–Tucker condition coincides with the strong stationary condition of the original problem. Furthermore, we propose a class of approximation methods to solve mathematical programs with second-order cone complementarity constraints. Any accumulation point of the iterative sequences, generated by the approximation method, is Clarke stationary under the corresponding linear independence constraint qualification. This stationarity can be enhanced to strong stationarity with an extra strict complementarity condition. Preliminary numerical experiments indicate that the proposed method is effective.

Relaxing Kink Qualifications and Proving Convergence Rates in Piecewise Smooth Optimization

In the paper [Optim. Methods Softw., 31 (2016), pp. 904--930] we derived first order (KKT) and second order (second order sufficiency condition (SOSC)) optimality conditions for functions defined by evaluation programs involving smooth elementals and absolute values. In that analysis, a key assumption on the local piecewise linearization was the linear independence kink qualification (LIKQ), a generalization of the linear independence constraint qualification (LICQ) known from smooth nonlinear optimization. Firstly, we show here that under LIKQ and SOSC with strict complementarity, the natural algorithm of successive abs-linear minimization with a proximal term (SALMIN) achieves a linear rate of convergence. A version of SALMIN called LiPsMin has already been implemented and tested in [Math. Program., 158 (2016), pp. 383--415; Math. Program. (2018), https://doi.org/10.1007/s10107-018-1273-5]. Secondly, we observe that, even without any kink qualifications, local optimality of the nonlinear objective always requires local optimality of its piecewise linearization, and strict minimality of the latter is in fact equivalent to sharp minimality of the former. Moreover, we show that SALMIN will converge quadratically to such sharp minimizers, where the function exhibits linear growth. These results are independent of the particular function representation, and allow in particular duplications of switching variables and other intermediates. Furthermore, we derive for nonsharp minimizers necessary and sufficient optimality conditions without any kink qualification. Finally, we present numerical results to illustrate the obtained convergence results.

Optimality conditions for vector optimization problem governed by the cone constrained generalized equations

ABSTRACTThe paper considers a class of vector optimization problems with cone constrained generalized equations. By virtue of advanced tools of variational analysis and generalized differentiation, a limiting normal cone of the graph of the normal cone constrained by the second-order cone is obtained. Based on the calmness condition, we derive an upper estimate of the coderivative for a composite set-valued mapping. The necessary optimality condition for the problem is established under the linear independent constraint qualification. As a special case, the obtained optimality condition is simplified with the help of strict complementarity relaxation conditions. The numerical results on different examples are given to illustrate the efficiency of the optimality conditions.

Whitney differentiability of optimal-value functions for bound-constrained convex programming problems

ABSTRACTIn the spirit of the Whitney Extension Theorem, consider a function on a compact subset of Euclidean space to be ‘Whitney-differentiable’ if it is a restriction of a continuously Fréchet-differentiable function with an open domain. Whitney-differentiable functions have been shown to have useful (yet possibly nonunique) derivatives and calculus properties even on the boundaries of their domains. This article shows that optimal-value functions for bound-constrained convex programmes with Whitney-differentiable objective functions are themselves Whitney-differentiable, even when the linear-independence constraint qualification is not satisfied. This result extends classic sensitivity results for convex programmes, and generalizes recent work. As an application, sufficient conditions are presented for generating continuously differentiable convex underestimators of nonconvex functions for use in methods for deterministic global optimization in the multivariate McCormick framework. In particular, the main result is applied to generate Whitney-differentiable convex underestimators for quotients of functions with known Whitney-differentiable relaxations.

Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow

Solutions to nonlinear, nonconvex optimization problems can fail to satisfy the KKT optimality conditions even when they are optimal. This is due to the fact that unless constraint qualifications (CQ) are satisfied, Lagrange multipliers may fail to exist. Even if the KKT conditions are applicable, the multipliers may not be unique. These possibilities also affect AC optimal power flow (OPF) problems which are routinely solved in power systems planning, scheduling and operations. The complex structure -- in particular the presence of the nonlinear power flow equations which naturally exhibit a structural degeneracy -- make any attempt to establish CQs for the entire class of problems very challenging. In this paper, we resort to tools from differential topology to show that for AC OPF problems in various contexts the linear independence constraint qualification is satisfied almost certainly, thus effectively obviating the usual assumption on CQs. Consequently, for any local optimizer there generically exists a unique set of multipliers that satisfy the KKT conditions.

Generalized Sensitivity Analysis of Nonlinear Programs

This paper extends classical sensitivity results for nonlinear programs to cases in which parametric perturbations cause changes in the active set. This is accomplished using lexicographic directional derivatives, a recently developed tool in nonsmooth analysis based on Nesterov's lexicographic differentiation. A nonsmooth implicit function theorem is augmented with generalized derivative information and applied to a standard nonsmooth reformulation of the parametric KKT system. It is shown that the sufficient conditions for this implicit function theorem variant are implied by a KKT point satisfying the linear independence constraint qualification and strong second-order sufficiency. Mirroring the classical theory, the resulting sensitivity system is a nonsmooth equation system which admits primal and dual sensitivities as its unique solution. Practically implementable algorithms are provided for calculating the nonsmooth sensitivity system's unique solution, which is then used to furnish B-subdifferenti...

Numerical Optimal Control With Periodicity Constraints in the Presence of Invariants

Periodic optimal control problems (POCPs) based on dynamic models holding invariants are often problematic to treat using standard numerical methods. The difficulty stems from a failure of standard constraint qualifications and typically hinders the convergence of the numerical solver, or even defeats it. Optimization problems having weak constraint qualifications can be treated using dedicated solvers, at the price of a more involved algorithmic. In this paper, we analyze the constraint qualification of POCPs holding invariants, and propose three simple and computationally inexpensive modifications of the formulation that allow for a recovery of linear independence constraint qualification, while not affecting the second-order sufficient conditions for optimality. Hence, the resulting POCP can be tackled via standard solvers, without special treatment. The application of these approaches is detailed for the case of POCPs holding index-reduced differential-algebraic equations and representations of the $\mathrm{SO}\left(3\right)$ Lie group.

Constraint Qualification Based Detection Method for Nodal Price Multiplicity

This paper proposes a fast, standalone method for detecting multiple dual solutions (MDS) of the market dispatch problem. Using linear independence constraint qualification conditions and matrix theory, we characterize the conditions for MDS and develop a procedure for their detection. Our procedure exploits the structure of the market dispatch problem to improve computational efficiency. We demonstrate our method on eight nodal pricing scenarios, in four test systems.

Sensitivity Analysis of Multiobjective Optimization Problems with Parameterized Quasi-Variational Inequalities

ABSTRACTThis paper mainly establishes the sensitivity analysis of a multiobjective optimization problem with parameterized quasi-variational inequalities (QVIs). Using the (regular) coderivative of the associated epigraphical multifunction, the (regular) subdifferentials of the efficient frontier maps are estimated, which involve the (regular) coderivatives of the solution mapping to the parameterized QVIs. Under the linear independent constraint qualification, the defined auxiliary set-valued mappings in the parameterized QVIs are clam. The detailed formulae of subdifferentials of the efficient frontier maps are obtained and examples are simultaneously provided for analyzing and illustrating the obtained results.

On M-stationarity conditions in MPECs and the associated qualification conditions

Depending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed qualification conditions as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible qualification conditions, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive the so-called M-stationarity conditions. The restrictiveness of assumptions and the strength of conclusions in the two forms of the MPEC is also strongly related to the qualification conditions on the “lower level”. For instance, even under the linear independence constraint qualification (LICQ) for a lower level feasible set described by $$\mathscr {C}^1$$ functions, the calmness properties of the original and the enhanced perturbation mapping are drastically different. When passing to $$\mathscr {C}^{1,1}$$ data, this difference still remains true under the weaker Mangasarian–Fromovitz constraint qualification, whereas under LICQ both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. After clarifying these relations, we provide a compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions. The obtained results are finally applied to MPECs with structured equilibria.

Generic Properties for Semialgebraic Programs

In this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints, in which every problem of the class is obtained by linear perturbations of the objective function, while the “core” objective function and the constraint functions are kept fixed. Assume that the linear independence constraint qualification is satisfied at every point in the constraint set. It is shown that almost all problems in the class are such that (i) the restriction of the objective function on the constraint set is coercive and regular at infinity; (ii) there is a unique optimal solution, lying on a unique active manifold, at which the strict complementarity and second order sufficiency conditions, the quadratic growth condition, and the Hölder type global error bound hold, and (iii) all minimizing sequences converge. Furthermore, the active manifold is constant, and the optimal solution and the optimal value function depend analytically under local perturbations of the objective function. These results are combined with a standard result about the existence of sums of squares certificates to prove that we can build a sequence of semidefinite programs whose solutions give rise to a sequence of points converging to the optimal solution of the original problem in finitely many steps. It is worth emphasizing that the results of this study hold globally and we do not require the problem to be convex or the constraint set to be compact.

Pointwise Constraints in Vector-Valued Sobolev Spaces

We consider a set \(\mathcal {C}\) with pointwise constraints in a vector-valued Sobolev space. We characterize its tangent and normal cone. Under the additional assumption that the pointwise constraints are affine and satisfy the linear independence constraint qualification, we show that the set \(\mathcal {C}\) is polyhedric. The results are applied to the optimal control of a string in a polyhedral tube.

First- and second-order optimality conditions for piecewise smooth objective functions

Any piecewise smooth function that is specified by an evaluation procedure involving smooth elemental functions and piecewise linear functions like and can be represented in the so-called abs-normal form. By an extension of algorithmic, or automatic, differentiation, one can then compute certain first- and second-order derivative vectors and matrices that represent a local piecewise linearization and provide additional curvature information. On the basis of these quantities, we characterize local optimality by first- and second-order necessary and sufficient conditions, which generalize the corresponding Kuhn-Tucker-Karush (KKT) theory for smooth problems. The key assumption is the linear independence kink qualification, a generalization of Linear Independence Constraint Qualification (LICQ) familiar from nonlinear optimization. It implies that the objective has locally a so-called decomposition and renders everything tractable in terms of matrix factorizations and other simple linear algebra operations. By yielding descent directions, whenever they are violated the new optimality conditions point the way to a superlinearly convergent generalized Quadratic Program solver, which is currently under development. We exemplify the theory on two nonsmooth examples of Nesterov.

Some new facts about sequential quadratic programming methods employing second derivatives

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.

Strong Stationarity for Optimization Problems with Complementarity Constraints in Absence of Polyhedricity

We consider mathematical programs with complementarity constraints in Banach spaces. In particular, we focus on the situation that the complementarity constraint is defined by a non-polyhedric cone K. We demonstrate how strong stationarity conditions can be obtained in an abstract setting. These conditions and their verification can be made more precise in the case that Z is a Hilbert space and if the projection onto K is directionally differentiable with a derivative as given in Haraux (Journal of the Mathematical Society of Japan 29(4), 615–631, 1977, Theorem 1). Finally, we apply the theory to optimization problems with semidefinite and second-order-cone complementarity constraints. We obtain that local minimizers are strongly stationary under a variant of the linear-independence constraint qualification, and these are novel results.