“Establishing a Group for Testing Mathematical Programs”

1982 ◽  
pp. 329-336
Author(s):  
John M. Mulvey
Keyword(s):  
Mathematical Programs

scholarly journals Topological Approach to Mathematical Programs with Switching Constraints

2021 ◽  
Author(s):  
Vladimir Shikhman
Keyword(s):  
Stationary Point ◽  
Mountain Pass Theorem ◽  
Strong Stability ◽  
Complete Characterization ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Linear Independence Constraint Qualification ◽  
Point Set ◽  
Stationary Level

AbstractWe study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

Plan, Projection, Policy—Mathematical Programming and Planning Theory

1974 ◽  
Vol 6 (4) ◽  
pp. 419-429 ◽  
Author(s):  
L D Hopkins
Keyword(s):  
Optimal Choice ◽  
Planning Theory ◽  
Public Agency ◽  
Indirect Determination ◽  
Mathematical Programs ◽  
Planning Models ◽  
Definition Of ◽  
The Individual ◽  
The Cost ◽  
Programming Analysis

The purpose of planning models is to help determine how best to change the controllable variables of a system so that the system will move toward desired goals. In the case of public planning, the welfare criterion or planning objective is not likely to be the same as the objective function driving the behavior of the individual elements of the system. A rigorous definition of a theory of planning is provided through extensions to the linear-programming analysis of the decentralized firm in order to handle (1) the distinction between the planning objectives and behavioral objectives, and (2) the indirect determination of prices by a public agency through policies which modify existing prices sufficiently to achieve a prescribed allocation of resources. A set of three related mathematical programs is identified which yield respectively a target plan, a projection, and an optimal choice of policies. A combination of these programs defines the best solution in terms of the value of the target plan minus the cost of altering the system to achieve that target.

Exact penalty results for mathematical programs with vanishing constraints

2010 ◽  
Vol 72 (5) ◽  
pp. 2514-2526 ◽  
Author(s):  
Tim Hoheisel ◽  
Christian Kanzow ◽  
Jiří V. Outrata
Keyword(s):  
Exact Penalty ◽  
Mathematical Programs ◽  
Vanishing Constraints
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