Convex Programming Algorithm Research Articles

Globally Convergent Algorithms for Convex Programming

We consider solving a (minimization) convex program by sequentially solving a (minimization) convex approximating subproblem and then executing a line search on an exact penalty function. Each subproblem is constructed from the current estimate of a solution of the given problem, possibly together with other information. Under mild conditions, solving the current subproblem generates a descent direction for the exact penalty function. Minimizing the exact penalty function along the current descent direction provides a new estimate of a solution, and a new subproblem is formed. For any arbitrary starting estimate, this scheme generates a sequence of estimates that converges to a solution of the given problem. Moreover, the functions defining the given problem and each subproblem need not be differentiable.

An approach to on-line power dispatch with ambient air pollution constraints

An on-line optimal environmental/economic dispatch methodology for electric power generation is developed in this paper. Aside from the conventional economic dispatch, constraints on air quality (such as those specified by the U.S. Environmental Protection Agency) are added to the minimum fuel cost problem. Using the integrated Gaussian puff model based on the statistical turbulent theory, rapid dynamic features of pollutant dispersion and its forecast surrounding the plants are emphasized. By applying a convex programming algorithm repeatedly, a set of marginal environmental imposts for the power plants at different times are obtained. Such imposts are incorporated with the fuel cost in the ordinary short-term economic dispatching program to indirectly account for the environmental impact of power generation on the quality of the ambient air. The approach is specifically taken to have little modification for existing economic dispatch programs and be implemented for real power networks. The proposed approach has been simulated in a power system with three plants and three monitoring points.

The Generalized Slack Variable Linear Program

The dilemma of when to cease analysis of a subproblem and go on to the next subproblem in convex programming is an especially difficult one. This paper views convex programming as an extension of linear programming and focuses attention on the analysis of a given subproblem. Specifically, if K = {x∣Ax ≤ b} is bounded with a nonempty interior and represents a residual unsearched region in some convex programming algorithm, what is (are) the “best” trial point(s) which can be selected? The generalized slack variable linear program (GSVLP) provides a mechanism for implementing trial point selection based on many strategies. Several implications of the L2 norm are revealed which are especially important in constraint deletion and rate of reduction of the hypervolume of the residual search region K.

Technical Note—Column Dropping in the Dantzig-Wolfe Convex Programming Algorithm: Computational Experience

This paper presents some computational experience on column dropping using the Dantzig-Wolfe convex programming algorithm. Computational results show virtually no difference in the measures of computational efficiency when all columns are retained and when only basic columns are retained.

Nested Decomposition of Multistage Convex Programs

The multistage or staircase structure appears naturally in many models with time horizons. This paper presents and discusses a method for decomposition when the problem functions are convex. Among the techniques which can be used to solve the subproblems are the Dantzig–Wolfe convex programming algorithm and Bender’s decomposition. Furthermore, when the nature of the problem presents certain structural forms, the decomposition allows for the introduction of more efficient techniques.

On fractional programming and equivalence

AbstractUnder fairly general conditions, a nonlinear fractional program, where the function to be maximized has the form f(x)/g(x), is shown to be equivalent to a nonlinear program not involving fractions. The latter program is not generally a convex program, but there is often a convex program equivalent to it, to which the known algorithms for convex programming may be applied. An application to duality of a fractional program is discussed.

Feasibility of Rent and Tax Incentives for Renovation in Older Residential Neighborhoods

Can city governments take meaningful actions to relieve the growing shortage of rental housing units for low income families without heavy reliance on state or federal support? To answer this question a net present value model is developed from the viewpoint of the existing landlords. Using this model the municipal policies which optimize the objective function of providing the maximum renovated housing units subject to budgetary constraints are identified by a convex programming algorithm. The empirical application of this approach to a Pittsburgh neighborhood indicates that it is possible to increase the quality and quantity of the rental housing stock while actually increasing the municipal net revenue.

Technical Note—Computational Experience with Normed and Nonnormed Column-Generation Procedures in Nonlinear Programming

Nemhauser and Widhelm have suggested a normed variant of the Dantzig-Wolfe convex programming algorithm. The normalization explicitly occurs in the multiplier-space. There the interior point used to generate the next cut is one that maximizes the minimum slack from the present cuts to that point. Nemhauser and Widhelm suggested this normalization for two reasons. First, since the corresponding cut at worst includes the generating point, choosing that point “centered” in some geometric sense should result in a generally efficient reduction in the feasible search region. Second, cuts not active in defining the present feasibility region can affect the choice of the interior-point if norming is not applied. This note demonstrates how primal feasible points can be generated at each iteration and cites limited computational results indicating that the normed procedure is generally more efficient than the original algorithm from the standpoint of both the number of iterations and computational time.

A METHOD FOR INCORPORATING AGRICULTURAL RISK INTO A WATER RESOURCE SYSTEM PLANNING MODEL1

ABSTRACT Prior studies of water resource systems have considered risk from the point of view that only the system planners could react to the effects of the risk elements. However, users of water from a system also react to risk. When the quantity of water that a system can supply is subject to considerable variation, the reactions of the users of the water will often effect the benefits generated by the system and thus its optimal design characteristics. A simulation model of a reservoir‐irrigation system is developed which incorporates the water users' reactions to risk in such a manner as to reflect their influence on the optimal design characteristics. A risk (convex) programming algorithm is incorporated into the model to reflect the water users' reactions to various levels of aversion to risk and degree of uncertainty in water deliveries. Response surfaces are generated as a result of performing the simulation at different levels of the design variables. An examination of these surfaces reveals the importance of including water users' reactions to risk in water resource system planning. The effects of different levels of risk aversion on the irrigation farmers' choice of crop enterprises are also examined.

Cutting plane algorithms and state space constrained linear optimal control problems

In this paper an algorithm is proposed for solving continuous linear optimal control systems with state space constraints by solving a sequence of linear optimal control systems without state space constraints. The convergence of the algorithm is proved by a method similar to cutting plane algorithm for convex programs in Banach Spaces. It is also shown how to solve the problem by using mathematical programming algorithm on the discretized problem. A numerical example is solved by discretization and mathematical programming.

On the distance between polytopes

This paper is written to fill an apparent gap in the literature in connection with some elementary problems of distance in finite dimensional normed linear spaces. In particular the problem of determining the distance between two polytopes in a normed n n -dimensional real vector space is considered. Special consideration is given to the case in which the norm is a twice differentiable function of its arguments and for this case a convex programming algorithm is presented. In addition several other cases are considered including the well known discrete Tchebycheff approximation. The results should find application in approximation theory.

An approximation technique for the optimal control of linear distributed parameter systems with bounded inputs

For a general model of a linear distributed parameter system, the problem of minimizing the norm of the difference between desired system response and obtainable system response is considered. Here the control input is constrained to be bounded in magnitude. An optimal solution is shown to exist, and an optimal solution in a class of controls dense in the constraint set is shown to exist. This latter class is characterized by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> parameters whose values are obtainable by a convex programming algorithm presented in the paper. The technique developed can also be directly applied to lumped parameter systems, lumped parameter driven distributed parameter systems and the optimum magnitude constrained input final value control problem for any of the preceding.

AN ALGORITHM FOR CONVEX PROGRAMMING

AN ALGORITHM FOR CONVEX PROGRAMMING