Dual Optimal Solutions Research Articles

TWO EXAMPLES IN A MARKET WITH TWO TYPES OF INDIVISIBLE GOOD

We consider an extension of the permutation game of Tijs et al (1984) in which players are endowed with and ultimately wish to consume one unit of each of two types of good (i.e., a house and a car). We present two examples. The first is a case where even though the corresponding linear program (CLP) does not solve with integers, the core of the market is not empty. The second example is a case with additively separable preferences in which there is a core vector in the market which does not correspond to any optimal dual solution of the CLP. Both examples demonstrate possible behavior that is impossible in many of the standard matching games.

Fairness versus efficiency in charging for the use of common facilities

The problem of efficiency vs fairness is considered in relation to the splitting of costs for shared facilities between users. This is considered as a result of a problem of sharing the cost of the provision of central computing facilities between different faculties in a large university, but the basic problem is widespread. A linear programming model is considered in order to minimise cost. The dual of this model is shown to correspond to an efficient allocation of costs. An alternative optimal dual solution is shown to give a ‘fair’ solution according to criteria resulting from cooperative game theory.

Primal and dual convergence of a proximal point exponential penalty method for linear programming

We consider the diagonal inexact proximal point iteration \(\frac{u^k-u^{k-1}}{\lambda_k}\in-\partial_{\varepsilon_k} f( u^k,r_k) + \nu^k\) where f(x,r)=cTx+r∑exp[(Aix-bi)/r] is the exponential penalty approximation of the linear program min{cTx:Ax≤b}. We prove that under an appropriate choice of the sequences λk, ek and with some control on the residual νk, for every rk→0+ the sequence uk converges towards an optimal point u∞ of the linear program. We also study the convergence of the associated dual sequence μik=exp[(Aiuk-bi)/rk] towards a dual optimal solution.

Subdifferentiability and the Duality Gap

We point out a connection between sensitivity analysis and the fundamental theorem of linear programming by characterizing when a linear programming problem has no duality gap. The main result is that the value function is subdifferentiable at the primal constraint if and only if there exists an optimal dual solution and there is no duality gap. To illustrate the subtlety of the condition, we extend Kretschmer's gap example to construct (as the value function of a linear programming problem) a convex function which is subdifferentiable at a point but is not continuous there. We also apply the theorem to the continuum version of the assignment model.

Entropic perturbation method for solving a system of linear inequalities

The problem of finding an x∈ R n such that Ax⩽ b and x⩾0 arises in numerous contexts. We propose a new optimization method for solving this feasibility problem. After converting Ax⩽ b into a system of equations by introducing a slack variable for each of the linear inequalities, the method imposes an entropy function over both the original and the slack variables as the objective function. The resulting entropy optimization problem is convex and has an unconstrained convex dual. If the system is consistent and has an interior solution, then a closed-form formula converts the dual optimal solution to the primal optimal solution, which is a feasible solution for the original system of linear inequalities. An algorithm based on the Newton method is proposed for solving the unconstrained dual problem. The proposed algorithm enjoys the global convergence property with a quadratic rate of local convergence. However, if the system is inconsistent, the unconstrained dual is shown to be unbounded. Moreover, the same algorithm can detect possible inconsistency of the system. Our numerical examples reveal the insensitivity of the number of iterations to both the size of the problem and the distance between the initial solution and the feasible region. The performance of the proposed algorithm is compared to that of the surrogate constraint algorithm recently developed by Yang and Murty. Our comparison indicates that the proposed method is particularly suitable when the number of constraints is larger than that of the variables and the initial solution is not close to the feasible region.

Existence and Determination of Competitive Equilibrium in Unit Commitment Power Pool Auctions: Price Setting and Scheduling Alternatives

The existence, determination, and effects of competitive market equilibrium for unit commitment power pool auctions are investigated in this paper. When an equilibrium does not exist, under specific situations conflictive multiple optimal primal solutions may exist. When an equilibrium exists, multiple primal solutions do not represent conflicts of interest. The existence or nonexistence of competitive equilibrium can be determined if the dual problem is solved to optimality. If equilibrium does not exist, there is excess supply at the optimal dual solution, which can be used to define priority orders and price-setting altematives to determine a final schedule, and avoid the conflicts of interest and revenue deficiency. Under disequilibrium, the optimal dual variables are not market clearing prices; a nonuniform pricing rule that avoids the flaws and complications of other pricing rules, such as maximum average cost and price minimization auctions, is proposed in the paper. The proposed scheduling and price-setting alternatives show that unit commitment models can be used in a market environment.

Validating absolute weight bounds in Data Envelopment Analysis (DEA) models

The Charnes, Cooper and Rhodes (CCR) DEA model and its linear forms maximise the efficiency of the assessed decision making unit (DMU) and, at the same time, the ratio of this efficiency to the maximum efficiency taken across all the DMUs, the latter naturally always being equal to one. It has been shown recently that, in the presence of absolute weight bounds, these models may not maximise the ratio of these efficiencies, a fact that may cause problems with the interpretation and use of the optimal primal and dual solutions. For example, an inefficient DMU may have greater efficiency than its target unit for some weights. This paper investigates the problem in greater detail; it shows that, in the linear DEA model maximising the total virtual output of the assessed DMU, the problem occurs only if upper bounds are imposed on the output weights. A similar result is established for the model that minimises the total virtual input.

Certificates of Primal or Dual Infeasibility in Linear Programming

In general if a linear program has an optimal solution, then a primal and dual optimal solution is a certificate of the solvable status. Furthermore, it is well known that in the solvable case, then the linear program always has an optimal basic solution. Similarly, when a linear program is primal or dual infeasible then by Farkas's Lemma a certificate of the infeasible status exists. However, in the primal or dual infeasible case then there is not an uniform definition of what a suitable basis certificate of the infeasible status is. In this work we present a definition of a basis certificate and develop a strongly polynomial algorithm which given a Farkas type certificate of infeasibility computes a basis certificate of infeasibility. This result is relevant for the recently developed interior-point methods because they do not compute a basis certificate of infeasibility in general. However, our result demonstrates that a basis certificate can be obtained at a moderate computational cost.

On the core of transportation games

The main purpose of this paper is to study the core of the so-called transportation games, which constitute an extension of the assignment games. We prove the nonemptiness of the core for these transportation games, and some results about the relationship between the core and the dual optimal solutions of the underlying transportation problem are also provided.

On improved Choi–Goldfarb solution-containing ellipsoids in linear programming

Ellipsoids that contain all optimal primal solutions, those that contain all optimal dual slack solutions, and primal–dual ellipsoids are derived. They are independent of the algorithm used and have a smaller size than the Choi–Goldfarb ellipsoids [J. Optim. Theory Appl. 80 (1994) 161–173].

The dual parameterization approach to optimal least square FIR filter design subject to maximum error constraints

This paper is concerned with the design of linear-phase finite impulse response (FIR) digital filters for which the weighted least square error is minimized, subject to maximum error constraints. The design problem is formulated as a semi-infinite quadratic optimization problem. Using a newly developed dual parameterization method in conjunction with the Caratheodory's dimensional theorem, an equivalent dual finite dimensional optimization problem is obtained. The connection between the primal and the dual problems is established. A computational procedure is devised for solving the dual finite dimensional optimization problem. The optimal solution to the primal problem can then be readily obtained from the dual optimal solution. For illustration, examples are solved using the proposed computational procedure.

A modified layered-step interior-point algorithm for linear programming

The layered-step interior-point algorithm was introduced by Vavasis and Ye. The algorithm accelerates the path following interior-point algorithm and its arithmetic complexity depends only on the coefficient matrixA. The main drawback of the algorithm is the use of an unknown big constant $$\bar \chi _A $$ in computing the search direction and to initiate the algorithm. We propose a modified layered-step interior-point algorithm which does not use the big constant in computing the search direction. The constant is required only for initialization when a well-centered feasible solution is not available, and it is not required if an upper bound on the norm of a primal—dual optimal solution is known in advance. The complexity of the simplified algorithm is the same as that of Vavasis and Ye. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Shadow Prices in Infinite-Dimensional Linear Programming

We consider the class of linear programs that can be formulated with infinitely many variables and constraints but where each constraint has only finitely many variables. This class includes virtually all infinite horizon planning problems modeled as infinite stage linear programs. Examples include infinite horizon production planning under time-varying demands and equipment replacement under technological change. We provide, under a regularity condition, conditions that are both necessary and sufficient for strong duality to hold. Moreover we show that, under these conditions, the Lagrangean function corresponding to any pair of primal and dual optimal solutions forms a linear support to the optimal value function, thus extending the shadow price interpretation of an optimal dual solution to the infinite dimensional case. We illustrate the theory through an application to production planning under time-varying demands and costs where strong duality is established.

An Efficient Algorithm for a Class of Two-Resource Allocation Problems

We propose an efficient algorithm for the two resource allocation problem defined on a series-parallel graph, where nodes represent tasks, and arcs represent precedence relationships. The completion time of each task is inversely proportional to the amount of resource allocated, and the objective is to minimize the length of the longest path on the graph. Our algorithm is derived based on a formal analysis of the primal and Lagrangian dual optimal solutions. The resulting search process always ends up with one of three exclusive cases. For the first two cases, a closed form solution exists. For the third case, a bisectional search is used. The computational complexity of the algorithm is thus bounded fromabove by O(Nlog(1/ɛ)), where N is the total number of tasks, and ∈ is any given accuracy.

Existence and duality theory for separated continuous linear programs

This paper surveys the recent developments in the theoretical study of separated continuous linear programs (SCLP). This problem serves as a useful model for various dynamic network problems where storage is permitted at the nodes. We demonstrate this by modelling some hypothetical problems of water distribution, transportation and telecommunications. The theoretical developments we present for SCLP fall into two main topics. The first of these is the existence of optimal solutions of various forms. These results culminate in one guaranteeing the existence of a piecewise analytic optimal solution, that is, having a finite number of breakpoints. The second topic we discuss is duality. Under this heading we develop a theory that closely resembles that for finite-dimensional linear programming. For instance, we define complementary slackness and give conditions under which there exist complementary slack primal and dual optimal solutions. Throughout the paper we observe that the main theorems are sufficiently general to include any reasonable practical problems

A Lagrangean Relaxation Scheme for Structured Linear Programs With Application To Multicommodity Network Flows

It is well known that linear programs can generally not be solved by straightforward Lagrangean relaxation and dual subgradient optimization, the reason being that the solutions to the Lagrangean relaxed problems are, normally, infeasible in the original linear program. This property is a consequence of the linearity of the problem and it holds even in the unlikely case that an exact optimal dual solution is found. We show that an optimal solution to the linear program can be obtained by calculating the simple average of the solutions to the relaxed problems which are solved during the subgradient search scheme, provided that the steplengths in this scheme are chosen according to a modified harmonic series. This method is similar to a procedure given earlier by Shor, which utilizes a particular weighted average and holds for any divergent series steplength rule. As an application of these two averaging schemes, we construct an approximate algorithm for the minimum cost multicommodity network flow problem. The algorithm has been implemented for solving multicommodity transportation problems and shows a good performance. The computational results indicate that it may be possible to use Lagrangean relaxation and dual subgradient optimization for constructing efficient special–purpose solution methods for structured large–scale linear programs

Perturbing the Dual Feasible Region for Solving Convex Quadratic Programs

A dual lp-norm perturbation approach is introduced for solving convex quadratic programming problems. The feasible region of the Lagrangian dual program is approximated by a proper subset that is defined by a single smooth convex constraint involving the lp-norm of a vector measure of constraint violation. It is shown that the perturbed dual program becomes the dual program as p→∞ and, under some standard conditions, the optimal solution of the perturbed dual program converges to a dual optimal solution. A closed-form formula that converts an optimal solution of the perturbed dual program into a feasible solution of the primal convex quadratic program is also provided. Such primal feasible solutions converge to an optimal primal solution as p→∞. The proposed approach generalizes the previously proposed primal perturbation approach with an entropic barrier function. Its theory specializes easily for linear programming.

Chapter 4 The lack of invariance of optimal dual solutions under translation

It is shown that optimal dual solutions for the Additive and the BCC models are not invariant under translation. In particular, this implies that the Ali-Seiford (1990) and Pastor (in chapter 3) results on translations apply only to the envelopment portion of DEA and thus do not, in particular, deal fully with domains having nonnegative inputs and outputs (but with no zero input or output vector).

Chapter 5 Duality, classification and slacks in DEA

This paper presents seven theorems which expand understanding of the theoretical structure of the Charnes-Cooper-Rhodes (CCR) model of Data Envelopment Analysis, especially with respect to slacks and the underlying structure of facets and faces. These theorems also serve as a basis for new algorithms which will provide optimal primal and dual solutions that satisfy the strong complementary slackness conditions (SCSC) for many (if not most) non-radially efficient DMUs; an improved procedure for identifying the setE of extreme efficient DMUs; and may, for many DEA domains, also settle in a single pass the existence or non-existence of input or output slacks in each of their DMUs. This paper also introduces the concept of a positivegoal vector G, which is applied to characterize the set of all possible maximal optimal slack vectors. The appendix C presents an example which illustrates the need for a new concept,face regular, which focuses on the role of convexity in the intersections of radial efficient facets with the efficient frontier FR. The same example also illustrates flaws in the popular “sum of the slacks” methodology.

Explicit solutions for interval semidefinite linear programs

We consider the special class of semidefinite linear programs (IVP) maximize traceCX subject to L⪯A(X)⪯U , where C, X, L, U are symmetric matrices, A is an (onto) linear operator, and ⪯ denotes the Löwner (positive semidefinite) partial order. We present explicit representations for the general primal and dual optimal solutions. This extends the results for standard linear programming that appeared in Ben-Israel and Charnes [3]. This work is further motivated by the explicit solutions for a different class of semidefinite problems presented recently in Yang and Vanderbei [15].