Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints

Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper, we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ, discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ.Communicated by Z. Q. Luo     

Nondegenerate necessary conditions for nonlinear optimal control problems with higher-index state constraints

For some optimal control problems with pathwise state constraints the standard versions of the necessary conditions of optimality are unable to provide useful information to select minimizers. There exist some literature on stronger forms of the maximum principle, the so-called nondegenerate necessary conditions, that can be informative for those problems. These conditions can be applied when certain constraint qualifications are satisfied. However, when the state constraints have higher index (i.e. their first derivative with respect to time does not depend on the control) these nondegenerate necessary conditions cannot be used. This happens because constraint qualifications assumptions are never satisfied for higher index state constraints. We note that control problems with higher index state constraints arise frequently in practice. An example is a common mechanical systems for which there is a constraint on the position (an obstacle in the path, for example) and the control acts as a second derivative of the position (a force or acceleration) which is a typical case. Here, we provide a nondegenerate form of the necessary conditions that can be applied to nonlinear problems with higher index state constraints. When addressing a problem with a state constraint of index k, the result described is applicable under a constraint qualification that involves the k -th derivative of the state constraint, corresponding to the first time when derivative depends explicitly on the control. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constraint Qualifications Characterizing Lagrangian Duality in Convex Optimization

In convex optimization, a constraint qualification (CQ) is an essential ingredient for the elegant and powerful duality theory. Various constraint qualifications which are sufficient for the Lagrangian duality have been given in the literature. In this paper, we present constraint qualifications which characterize completely the Lagrangian duality.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

A proximal method for identifying active manifolds

The minimization of an objective function over a constraint set can often be simplified if the “active manifold” of the constraints set can be correctly identified. In this work we present a simple subproblem, which can be used inside of any (convergent) optimization algorithm, that will identify the active manifold of a “prox-regular partly smooth” constraint set in a finite number of iterations.     

The p-Factor-Lagrange Methods for Degenerate Nonlinear Programming

The paper presents a new approach to solving nonlinear programming (NLP) problems for which the strict complementarity condition (SCC), a constraint qualification (CQ), and a second-order sufficient condition (SOSC) for optimality are not necessarily satisfied at a solution. Our approach is based on the construction of p-regularity and on reformulating the inequality constraints as equalities. Namely, by introducing the slack variables, we get the equality constrained problem, for which the Lagrange optimality system is singular at the solution of the NLP problem in the case of the violation of the CQs, SCC and/or SOSC. To overcome the difficulty of singularity, we propose the p-factor method for solving the Lagrange system. The method has a superlinear rate of convergence under a mild assumption. We show that our assumption is always satisfied under a standard second-order sufficient condition (SOSC) for optimality. At the same time, we give examples of the problems where the SOSC does not hold, but our assumption is satisfied. Moreover, no estimation of the set of active constraints is required. The proposed approach can be applied to a variety of problems.     

Strong duality and minimal representations for cone optimization

The elegant theoretical results for strong duality and strict complementarity for linear programming, LP, lie behind the success of current algorithms. In addition, preprocessing is an essential step for efficiency in both simplex type and interior-point methods. However, the theory and preprocessing techniques can fail for cone programming over nonpolyhedral cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, CQ, and strict complementarity, for linear cone optimization problems in finite dimensions. One theme is the notion of minimal representation of the cone and the constraints. This provides a framework for preprocessing cone optimization problems in order to avoid both the theoretical and numerical difficulties that arise due to the (near) loss of the strong CQ, strict feasibility. We include results and examples on the surprising theoretical connection between duality gaps in the original primal-dual pair and lack of strict complementarity in their homogeneous counterpart. Our emphasis is on results that deal with Semidefinite Programming, SDP.     

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments     

Probabilistic inequality constraints in stochastic optimal control theory

There are many optimal control problems in which it is necessary or desirable to constrain the range of values of state variables. When stochastic inputs are involved, these inequality constraint problems are particularly difficult. Sometimes the constraints must be modeled as hard constraints which can never be violated, and other times it is more natural to prescribe a probability that the constraints will not be violated. This paper treats general problems of the latter type, in which probabilistic inequality constraints are imposed on the state variables or on combinations of state and control variables. A related class of problems in which the state is required to reach a target set with a prescribed probability is handled by the same methods. It is shown that the solutions to these problems can be obtained by solving a comparatively simple bilinear deterministic control problem.     

Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification

A global error bound is given on the distance between an arbitrary point in then-dimensional real spaceR ⁿ and its projection on a nonempty convex set determined bym convex, possibly nondifferentiable, inequalities. The bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong Slater constraint qualification (CQ) which implies the ordinary Slater CQ. A very simple bound on the distance to the projection relative to the distance to a point satisfying the ordinary Slater CQ is given first and then used to derive the principal global error bound. This material is based on research supported by National Science Foundation Grant CCR-9322479 and Air Force Office of Scientific Research grant F49620-97-1-0326.     

A new approach to the incorporation of servo constraints in multibody dynamics

A new index reduction approach is developed to solve the servo constraint problems [2] in the inverse dynamics simulation of underactuated mechanical systems. The servo constraint problem of underactuated systems is governed by differential algebraic equations (DAEs) with high index. The underlying equations of motion contain both holonomic constraints and servo constraints in which desired outputs (specified in time) are described in terms of state variables. The realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Since the (differentiation) index of the DAEs is often higher than three for underactuated systems, in which the number of degrees of freedom is greater than the control outputs/inputs, we propose a new index reduction method [1] which makes possible the stable numerical integration of the DAEs. We apply the proposed method to differentially flat systems, such as cranes [1,4,5], and non-flat underactuated systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical method for the servo constraint problem of underactuated mechanical systems

Servo constraints are used in inverse dynamics simulations of discrete mechanical systems, especially for trajectory tracking control problems [1], whose desired outputs are represented by state variables and treated as servo constraints [2]. Servo constraint problems can be classified into fully actuated and underactuated multibody systems, and the equations of motion take the form of differential algebraic equations (DAEs) including holonomic and servo constraints. For fully actuated systems, control inputs can be solved from the equations by model inversion, as the input distribution matrix is nonsingular and invertible. However, underactuated systems have more degrees of freedom than control inputs. The input distribution matrix is not invertible, and in contrast to passive constraints, the realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Therefore, it is challenging for the determination of control inputs which force the underactuated system to realize the partly specified motion. For differentially flat underactuated systems, the differentiation index of DAEs may exceed three. Hence we need to apply specific index reduction techniques, such as the projection approach applied in [3], [4], and [6]. The present work applies index reduction by minimal extension [5] to differentially flat underactuated crane systems and shows that the index can be reduced from five to three and even to one. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hierarchical production policies in stochastic two-machine flowshops with finite buffers

This paper concerns production planning in manufacturing systems with two unreliable machines in tandem. The problem is formulated as a stochastic control problem in which the objective is to minimize the expected total cost of production, inventories, and backlogs. Since the sizes of the internal and external buffers are finite, the problem is one with state constraints. As the optimal solutions to this problem are extremely difficult to obtain due to the uncertainty in machine capacities as well as the presence of state constraints, a deterministic limting problem in which the stochastic machine capacities are replaced by their mean capacities is considered instead. The weak Lipschitz property of the value functions for the original and limiting problems is introduced and proved; a constraint domain approximation approach is developed to show that the value function of the original problem converges to that of the limiting problem as the rate of change in machine states approaches infinity. Asymptotic optimal production policies for the orginal problem are constructed explicity from the near-optimal policies of the limiting problem, and the error estimate for the policies constructed is obtained. Algorithms for constructing these policies are presented.This work was partly supported by CUHK Direct Grant 220500660, RGC Earmarked Grant CUHK 249/94E, and RGC Earmarked Grant CUHK 489/95E.     

均衡约束数学规划问题的一种新的约束规格

本文研究了均衡约束数学规划(MPEC)问题.利用其弱稳定点,获得了一种新的约束规格–MPEC的伪正规约束规格.用一种简单的方式,证明了该约束规格是介于MPEC-MFCQ(即MPEC,Mangasarian-Fromowitz约束规格)与MPEC-ACQ(即MPEC,Abadie约束规格)之间的约束规格,因此该约束规格也可以导出MPEC问题的M-稳定点.最后通过两个例子,说明了该约束规格与MPEC-MFCQ以及与MPEC-ACQ之间是严格的强弱关系.     

Hidden conic quadratic representation of some nonconvex quadratic optimization problems

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.     

Necessary Optimality Conditions for Optimal Control Problems with Nonsmooth Mixed State and Control Constraints

In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition and under the assumption that the state and control constraint functions are smooth. In this paper we derive necessary optimality conditions for problems with nonsmooth mixed state and control constraints under constraint qualifications based on pseudo-Lipschitz continuity and calmness of certain set-valued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. Moreover necessary optimality conditions with an Euler inclusion taking an explicit multiplier form are derived for certain cases.     

A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.     

Bounded knapsack sharing

A bounded knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more linear inequality constraints, an objective function composed of single variable continuous functions called tradeoff functions, and lower and upper bounds on the variables. A single constraint problem which can have negative or positive constraint coefficients and any type of continuous tradeoff functions (including multi-modal, multiple-valued and staircase functions) is considered first. Limiting conditions where the optimal value of a variable may be plus or minus infinity are explicitly considered. A preprocessor procedure to transform any single constraint problem to a finite form problem (an optimal feasible solution exists with finite variable values) is developed. Optimality conditions and three algorithms are then developed for the finite form problem. For piecewise linear tradeoff functions, the preprocessor and algorithms are polynomially bounded. The preprocessor is then modified to handle bounded knapsack sharing problems with multiple constraints. An optimality condition and algorithm is developed for the multiple constraint finite form problem. For multiple constraints, the time needed for the multiple constraint finite form algorithm is the time needed to solve a single constraint finite form problem multiplied by the number of constraints. Some multiple constraint problems cannot be transformed to multiple constraint finite form problems.     

Mathematical programs with vanishing constraints: a new regularization approach with strong convergence properties

Motivated by a recent method introduced by Kanzow and Schwartz [C. Kanzow and A. Schwartz, A new regularization method for mathematical programs with complementarity constraints with strong convergence properties, Preprint 296, Institute of Mathematics, University of Würzburg, Würzburg, 2010] for mathematical programs with complementarity constraints (MPCCs), we present a related regularization scheme for the solution of mathematical programs with vanishing constraints (MPVCs). This new regularization method has stronger convergence properties than the existing ones. In particular, it is shown that every limit point is at least M-stationary under a linear independence-type constraint qualification. If, in addition, an asymptotic weak nondegeneracy assumption holds, the limit point is shown to be S-stationary. Second-order conditions are not needed to obtain these results. Furthermore, some results are given which state that the regularized subproblems satisfy suitable standard constraint qualifications such that the existing software can be applied to these regularized problems.