Sufficient Conditions For Local Optimality Research Articles

Augmented Lagrangians and hidden convexity in sufficient conditions for local optimality

Augmented Lagrangians and hidden convexity in sufficient conditions for local optimality

Effective heuristic for large-scale unrelated parallel machines scheduling problems

This paper is concerned with non-preemptive scheduling of large-scale unrelated parallel machines (UPM) with the objective of minimizing total weighted completion times (TWCT). We propose a sequential improvement local search algorithm using multiple-jump strategy embedded within Tabu search (TS) components for TWCT, and use a highly efficient data structure to provide a necessary and sufficient condition for local optimality of a solution. We will generate a set of large-scale test problems to evaluate the performance of proposed algorithm in term of scalability, solution quality and efficiency. The non-parametric tests of algorithm components will be used to validate the consistent performance across problem types and problem sizes in the proposed algorithm.

Reduction of Sufficient Conditions for Optimal Control Problems With Subgroup Symmetry

In this paper, we apply symmetry reduction techniques from geometric mechanics to sufficient conditions for local optimality in optimal control problems. After reinterpreting some previous results for left-invariant problems on Lie groups, we focus on optimal control problems with subgroup symmetry. For these problems, the necessary conditions for optimality can be simplified by exploiting symmetries so as to reduce the number of variables needed to describe trajectories of the system. We show that sufficient conditions for optimality, based on the non-existence of conjugate points, can be simplified in an analogous way to the necessary conditions. We demonstrate these simplifications by analyzing an optimal control problem that models a spinning top in a gravitational field, and we give particular attention to the example of an axisymmetric sleeping top. The results we derive in this paper allow us to determine which trajectories of a sleeping top are locally optimal solutions of the optimal control problem, which is a new result that has not appeared in previous literature.

First‐order and second‐order sensitivity analyses for a body with a thin rigid inclusion

The paper deals with the two‐dimensional problem of the elastic equilibrium of a body containing a thin rigid inclusion. The rigid inclusion is considered to be soldered into elastic medium. The volume forces act on the body; at the external boundary the body is fixed. The first and second variations of the solution with respect to the shape of the domain are calculated. As an illustrative application of the results, the optimal shape design problem is considered. The necessary and sufficient conditions for local optimality are written. Copyright © 2014 John Wiley & Sons, Ltd.

Sufficiency and singularity in optimal control

optimization problem in a Banach space. The assumption that the strengthened Legendre– Clebsch condition holds is crucial. In particular, we refer the reader to Milyutin & Osmolovskii (1998) where the importance of this condition is fully explained. In this paper, we consider a fixed-endpoint optimal control problem with equality control constraints and provide a new set of sufficient conditions for a strong minimum that include the Legendre–Clebsch necessary condition but not its strengthened form. The sufficiency theorem includes also a new version of the strengthened Weierstrass condition. The proof is strongly related to a direct sufficiency proof (in the sense that it deals explicitly with the positivity of the second variation along admissible variations) given by Hestenes (1966) for the fixed-endpoint isoperimetric problem in the calculus of variations. The development of this technique as it appears in Hestenes (1966), as well as its application to more general problems, can be traced back to different papers of the author and McShane (see Hestenes, 1934, 1936, 1937, 1946, 1947a,b, 1948; McShane, 1942). The original proof deals with the concept, introduced by Hestenes, of a directional convergent sequence of trajectories (absolutely continuous functions) which is in turn a generalization of the concept of directional convergence for vectors in the finite-dimensional case. This notion relies on the specific calculus of variations problem considered in Hestenes (1966) but, as we show in this paper, it can be modified to cover optimal control problems. The modification we introduce not only shows that the main ideas of the proof can be generalized, but also provides a new set of sufficient conditions which, due to the possible singularity of the process under consideration, lies beyond the scope of the references mentioned above. It is worth mentioning that this technique has recently been extended in order to solve isoperimetric control problems without the requirement of transforming the original problem into a problem of Lagrange (see Rosenblueth & Sanchez Licea, 2011a). Previous attempts to generalize the proof of Hestenes can be found in Rosenblueth (1999) and Rosenblueth & Sanchez Licea (2010). In the first of these papers, we obtained sufficient conditions for local optimality, both weak and strong, assuming the strengthened Legendre–Clebsch condition. In the second one, several auxiliary results allowed us to establish the sufficiency theorem of Rosenblueth (1999) for weak minima in a new and clearer way. Those auxiliary results partly coincide with those used in the sufficiency proof for strong minima derived in this paper, without the assumption of the strengthened Legendre–Clebsch condition. The paper is organized as follows. In Section 2, we pose the problem we shall deal with, introduce some notation and basic definitions and state the main result. For comparison reasons we also state, as a corollary, the sufficiency theorem for strong minima obtained in Rosenblueth (1999). In Section 3, we explain certain relevant issues related to the new set of sufficient conditions with the idea of clarifying some of the underlying principles on which the proof is based. Section 4 is devoted to the proof of the sufficiency theorem together with the statement of some auxiliary results which are used throughout the proof. The auxiliary results, which deal with a generalization of the notion of a directional convergent sequence of trajectories, are established in Section 5. In the final section, we illustrate the usefulness of the new sufficiency theorem by means of six examples of singular processes which are strict local minima of the problems in hand. 2. Statement of the problem and the main results The fixed-endpoint optimal control problem we shall study in this paper can be stated as follows. Suppose we are given an interval T := [t0, t1] in R, two points ξ0 and ξ1 in R and functions L and f mapping T × R × R to R and R, respectively, and ψ mapping R to R. at U nivrsidad N aconal A tA 3om a de M A © xico on A ril 9, 2013 htp://im am ci.oxfournals.org/ D ow nladed from SUFFICIENCY AND SINGULARITY IN OPTIMAL CONTROL 39 Let X := AC(T ; R) denote the space of absolutely continuous functions mapping T to R, let U := L1(T ; R), set Z := X × U , and denote by Ze the set of all (x, u) ∈ Z satisfying (a) L(t, x(t), u(t)) is integrable on T . (b) ẋ(t)= f (t, x(t), u(t)) a.e. in T . (c) x(t0)= ξ0, x(t1)= ξ1. (d) ψ(u(t))= 0 (t ∈ T). The problem we shall deal with, which we label (P), is that of minimizing I over Ze, where

Perfect, strongly eutactic lattices are periodic extreme

We introduce a parameter space for periodic point sets, given as unions of m translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension d ⩽ 8 and d = 24 .

On the Subdifferentiability of the Difference of Two Functions and Local Minimization

We set up a formula for the Fréchet and ε-Fréchet subdifferentials of the difference of two convex functions. We even extend it to the difference of two approximately starshaped functions. As a consequence of this formula, we give necessary and sufficient conditions for local optimality in nonconvex optimization. Our analysis relies on the notion of gap continuity of multivalued maps and involves concepts of independent interest such as the notions of blunt and sharp minimizers and the notion of equi-subdifferentiability.

Second-Order Optimality Conditions for Constrained Domain Optimization

This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ? Ω g(x)?dx=1, a domain is sought which maximizes either $\mathcal{F}_{x_{0}}(\Omega )=F(x_{0},u(x_{0}))$ , fixed x 0?Ω, or ?(Ω)=? Ω F(x,u(x))?dx, where u solves the Dirichlet problem Δu(x)=?f(x), x?Ω, u(x)=0, x??Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals $\mathcal{F}_{x_{0}}$ and ?. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω.

Fault detection in a mixed H/sub 2//H/sub /spl infin// setting

In this note, we study the problem of fault detection in linear time-invariant (LTI) systems which are simultaneously effected by two classes of unknown inputs: Noises having fixed spectral densities and unknown finite energy disturbances. This problem is formulated as a mixed H/sub 2//H/sub /spl infin// filtering problem. Necessary and sufficient conditions for local optimality are presented. Moreover, it is shown that suboptimal solutions can be computed by solving a convex minimization problem with a set of linear matrix inequality (LMI) constrains. A numerical example is given to illustrate the advantage of the mixed H/sub 2//H/sub /spl infin// approach as compared to existing techniques which are based on optimization of H/sub 2/ and H/sub /spl infin// criteria.

A new dominance rule to minimize total weighted tardiness with unequal release dates

We present a new dominance rule by considering the time-dependent orderings between each pair of jobs for the single machine total weighted tardiness problem with release dates. The proposed dominance rule provides a sufficient condition for local optimality. Therefore, if any sequence violates the dominance rule then switching the violating jobs either lowers the total weighted tardiness or leaves it unchanged. We introduce an algorithm based on the dominance rule, which is compared to a number of competing heuristics for a set of randomly generated problems. Our computational results indicate that the proposed algorithm dominates the competing algorithms in all runs, therefore it can improve the upper bounding scheme in any enumerative algorithm. The proposed time-dependent local dominance rule is also implemented in two local search algorithms to guide these algorithms to the areas that will most likely contain the good solutions.

Second Order Sufficient Optimality Conditions for Some State-constrained Control Problems of Semilinear Elliptic Equations

This paper deals with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition. The case of boundary control is studied. Pointwise constraints on the control and certain equality and set-constraints on the state are considered. Second order sufficient conditions for local optimality of controls are established.

A new dominance rule for the total weighted tardiness problem

We present a new dominance rule for the single machine total weighted tardiness problem with job dependent penalties. The proposed dominance rule provides a sufficient condition for local optimality. We show that if any sequence violates the dominance rule, then switching the violating jobs either lowers the total weighted tardiness or leaves it unchanged. We also develop a new algorithm based on the dominance rule, which is compared to a number of competing heuristics for a set of randomly generated problems. Our computational results of over 40000 problems indicate that the proposed algorithm dominates the competing heuristics in all runs.

Optimal Control Problems with Partially Polyhedric Constraints

This paper discusses a class of state constrained optimal control problems, for which it is possible to formulate second-order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints. This kind of result is classical in the case of polyhedric control constraints. Our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric, in the presence of state constraints satisfying some specific hypotheses. The analysis is based on the assumption that some strict semilinearized qualification condition is satisfied. We apply the theory to some optimal control problems of elliptic equations with state and control constraints.

A new lower bounding scheme for the total weighted tardiness problem

We propose a new dominance rule that provides a sufficient condition for local optimality for the 1‖∑ w i T i problem. We prove that if any sequence violates the proposed dominance rule, then switching the violating jobs either lowers the total weighted tardiness or leaves it unchanged. Therefore, it can be used in reducing the number of alternatives for finding the optimal solution in any exact approach. We introduce an algorithm based on the dominance rule, which is compared to a number of competing approaches for a set of randomly generated problems. We also test the impact of the dominance rule on different lower bounding schemes. Our computational results over 30,000 problems indicate that the amount of improvement is statistically significant for both upper and lower bounding schemes.

Second order sufficient optimality conditions for a class of elliptic control problems

In this paper we deal with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition. The case of a boundary control is studied. We consider pointwise constraints on the control and certain equality and inequality constraints on the state. Second order sufficient conditions for local optimality of controls are derived.

Partially-linear transportation problems, obtained by means of absolute value functions

Nonlinear transportation problems may be successfully used to model problems in economics, due to the nonlinear relationship between quantities and the real cost for their transportation. In this paper we consider transportation problems with two-piece linear functions on arcs (which may be nonconvex), where the nonlinearity is represented by absolute value functions. A necessary and sufficient condition for local optimality is given and an algorithm for solving these problems is suggested.

Partially‐Linear Transportation Problems, Obtained by Means of Absolute Value Functions

Nonlinear transportation problems may be successfully used to model problems in economics, due to the nonlinear relationship between quantities and the real cost for their transportation. In this paper we consider transportation problems with two‐piece linear functions on arcs (which may be nonconvex), where the nonlinearity is represented by absolute value functions. A necessary and sufficient condition for local optimality is given and an algorithm for solving these problems is suggested.

Partially-linear transportation problems

The nonlinear relationship between quantities and the real cost for their transportation causes interest in some nonlinear transportation problems. Partially-linear transportation problems may be successfully applied in economics because every nonlinear function may be approximated by partially-linear functions. In this work we consider transportation problems with partially-linear dependence of the transportation costs and give the necessary and sufficient conditions for local optimality. A constructive method for solving such problems is suggested.

Second-Order Optimality Conditions in Sets of ${\text{L}}^\infty $ Functions with Range in a Polyhedron

Formal extensions of the general second-order necessary conditions and sufficient conditions for local optimality in a polyhedral convex set $U \subset \mathbb{R}^m $ are established for ${\text{L}}^\infty $-local optimality and ${\text{L}}^2 $-local optimality in the infinite-dimensional nonpolyhedral convex set $\Omega$ of ${\text{L}}^\infty $ functions $u( \cdot ):[0,1] \to U$. A more refined analysis for nonconvex cost functions with specially structured differentials yields optimality conditions that apply to an important class of constrained input Bolza optimal control problems. The gap between the necessary conditions and sufficient conditions in this setting is uncharacteristically small for infinite-dimensional problems. In the control problem context, the ${\text{L}}^\infty $-local optimality conditions and ${\text{L}}^2 $-local optimality conditions entail a mild strengthening of a pointwise strict complementarily condition and variants of the Legendre–Clebsch condition and the Pontrya.gin minimum principle. In related recent studies, similar second-order sufficient conditions for the special case $U = [0,\infty )$ are the key hypotheses in corresponding local convergence theories for iterative constrained minimization algorithms.

Algorithm for solving partially - linear optimization problems obtained by getting maximum of functions

The partially-linear models can be successfully applied in economics and technology, which increases the interest in them. Every continuous partially-linear function with a finite number of linear parts can be presented by means of linear functions and maximum of the functions. We considear graph interpretation of this optimization problem. Partially-linear problems are commonly not only nondifferentiable but nonconvex, which makes their solution by standard optimization methods difficult. In this work we give the necessary and sufficient conditions for local optimality. An algorithm for solving such problems, based on the constructive optimization methods is suggested