Feasible SetEdit
A feasible set is the collection of all decisions or actions that satisfy a given set of constraints in an optimization problem. It defines the universe of possibilities within which an objective is to be optimized and, in practice, encodes the real-world limits that shape choices—such as budgets, capacities, laws, contracts, and technology. In mathematical terms, if the decision vector x lies in a space X and is constrained by a set of inequalities and equalities, the feasible set is the subset of X that satisfies all those constraints. This notion is central to the way markets, firms, and policymakers think about what can be done, and it underpins many standard tools in optimization and constraint (mathematical) theory.
In most applied settings, the feasible set is the domain over which a decision maker searches for the best outcome. If the objective is to maximize profit, minimize risk, or improve welfare, the feasible set determines what is attainable before the best possible result can be reached. When constraints are well aligned with actual limits—such as production capacity, budget ceilings, or regulatory requirements—the feasible set offers a faithful map of real options. When constraints are overly loose or poorly specified, the feasible set may overstate what is possible; when constraints are too tight, it may obscure viable strategies. The balance between constraint realism and analytical tractability is a constant feature of modeling in production, finance, and public policy. For related geometric intuition, think of the feasible set as a region in space where every point corresponds to a feasible configuration of resources and decisions, much like the polyhedral regions that arise in linear programming and convex optimization.
Definition
Let X be the decision space, typically a subset of a Euclidean space such as ℝ^n. The feasible set F is defined by the constraints that decisions must satisfy: - inequality constraints: g_i(x) ≤ 0 for i = 1, ..., m - equality constraints: h_j(x) = 0 for j = 1, ..., p - domain restrictions: x ∈ D (often x ≥ 0 or similar)
A decision vector x is feasible if it lies in F, and infeasible otherwise. The objective function f(x) is then optimized over x ∈ F, yielding a solution that is sensitive to the shape and size of F. In many standard problems, F takes a particularly tractable form: a convex set (for instance, a polyhedron in linear programming) or a closed and bounded region in simple cases. When F is nonempty, optimization proceeds inside this set; when F is empty, the problem is infeasible because no decision satisfies all constraints. For a quick anchor, the classic linear program has F = { x : Ax ≤ b, x ≥ 0 }, a convex polyhedron in ℝ^n.
In economic and policy contexts, the feasible set often embodies the practical limits that govern choices. Constraints can encode technology (production possibility), finance (budget or capital limits), logistics (delivery capacity), and law (regulatory requirements). In public policy modeling, an additional layer—often described as political feasibility—can interact with the technical feasibility captured by F, shaping the space of options that lawmakers can realistically implement.
Examples
Linear programming: Maximize c^T x subject to Ax ≤ b, x ≥ 0. Here F is the polyhedron formed by the linear inequalities, and the optimum is found at a boundary point or a vertex of F in many cases. See linear programming for the standard framework, polyhedron for the geometric interpretation, and convex optimization for broader methods.
Knapsack problem: Choose items to maximize value subject to a weight limit. The feasible set consists of all item subsets whose total weight stays within the capacity; in many formulations, x is binary and F is discrete, reflecting the combinatorial nature of the problem.
Portfolio optimization: Select asset weights w to maximize expected return subject to risk constraints and budget constraints. The feasible set encodes diversification limits, leverage restrictions, and budget neutrality, linking finance portfolio optimization to the geometry of F.
Production planning under capacity: A factory must decide how much to produce of several products given resource constraints. F captures the nonnegative production levels that respect time, material, and labor limits.
Properties and structure
Convexity: If all constraints are convex (including linear ones) and there are no integer requirements, F is a convex set. This is especially important because it guarantees that any local optimum is a global optimum, enabling efficient solution methods in convex optimization.
Non-emptiness and emptiness: F is nonempty when feasible solutions exist; this is the prerequisite for any meaningful optimization. An empty feasible set signals infeasibility and often prompts reformulation of constraints or relaxation of requirements.
Boundedness and unboundedness: F can be bounded or unbounded. Unbounded feasible sets can lead to unbounded objective values in the absence of additional constraints or may indicate the need for more realistic limits in the model.
Closedness and openness: The topological properties of F (whether it is closed, open, or neither) influence the existence of optimum points, duality properties, and the behavior of numerical algorithms.
Relationship to the feasible region: In many contexts, especially in economics and operations research, the feasible set is referred to as a feasible region. While terminology varies, the core idea is the same: a region in decision space where all constraints are satisfied.
Applications and implications
In optimization practice, the feasible set is the primary stage on which all algorithms operate. The geometry of F affects the choice of method, from simple projected gradient steps to interior-point methods for large-scale convex problems. See feasibility problem and projection in algorithmic contexts for practical approaches to staying within F.
In economics and business strategy, the feasible set translates theory into action. Production plans, investment portfolios, and policy proposals are all evaluated within the constraints modeled in F. The quality of the results hinges on how well the constraints reflect reality and objectives. Concepts such as Pareto efficiency and economic efficiency often presuppose that feasible sets codify the relevant trade-offs cleanly enough to draw meaningful conclusions about improvement.
In public policy and regulatory design, expanding or tightening the feasible set corresponds to changes in law, incentives, or institutions. When policymakers adjust constraints, they alter F and thus the space of achievable outcomes. The tension between efficiency and other aims (fairness, safety, national security) often centers on how these constraints should be shaped, implemented, and enforced. See policy design and regulation for related perspectives.
Computation and complexity: Checking feasibility (whether a given x lies in F) and enumerating or optimizing over F are core computational tasks. For linear and convex problems, powerful algorithms exploit the structure of F; for nonconvex or discrete constraints, feasibility often becomes the hard part of the problem, requiring specialized techniques such as branch-and-bound or heuristic search.
Controversies and debates
From a market-oriented perspective, a central debate concerns how much and which constraints should be encoded in the feasible set. Proponents argue that a well-chosen, accurate set of constraints yields efficient, transparent decision rules that reflect real limits and encourage innovation within those limits. Critics may claim that models impose arbitrary or politically convenient constraints, potentially stifling useful experimentation or rapid response. In this view, the feasible set is not a neutral map but a mirror of institutional choices, and changes to F can have distributional consequences.
Some critics of abstract modeling argue that excessive reliance on a neatly defined feasible set can obscure important real-world frictions, such as information asymmetries, behavioral deviations, or dynamic resource depletion. Advocates counter that the feasible set is a necessary abstraction that helps decision makers compare options systematically; the solution is not to abandon the tool, but to calibrate it with accountable assumptions and timely data.
Woke criticisms that label mathematical modeling as inherently biased or discriminatory often focus on the allocation of constraints that affect outcomes across groups. A pragmatic counterpoint from this viewpoint is that constraints should be chosen to reflect objective limits and policy goals, not to reflect generosity toward a preferred outcome. In this frame, the critique is aimed at process and transparency rather than erasing the tool; the remedy is more explicit modeling choices, sensitivity analysis, and open discussion about which constraints are justified by technology, law, and economic efficiency, not a blanket rejection of optimization as such.