Chance-constrained portfolio selection¶

A portfolio optimization model that maximizes a return objective while limiting the probability that final wealth or another outcome falls below a declared safety threshold.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3646
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Financial Optimization → Economics & Finance

Core Idea¶

The probability guarantee is only as reliable as the return distribution, horizon and dependence model, and loss aversion, VaR-like constraints, transaction costs and estimation uncertainty alter the feasible set.^[1] Asset weights determine a random terminal outcome, a chance constraint restricts its lower-tail failure probability and an optimizer selects a feasible allocation with the best declared expected utility or return criterion. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of financial optimization. It is the domain-specific identity fixed by the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test. This gives the entry an operational identity rather than merely a historical label.

A useful analysis keeps three layers separate. The constitutive layer says what must be true: the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit. The evidential layer asks what observation or proof warrants the claim: type the carrier, state every parameter and convention in the definition, test that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases. The use layer asks what reasoning becomes available once the identity is established: recognizing and comparing instances of Chance-constrained portfolio selection, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Conflating the layers is the most common source of scope inflation.

How would you explain it like I'm…

Safe-Enough Money Split

Imagine you're choosing how to split your money between piggy banks that might grow or shrink. You want it to grow as much as you can, but you also make a rule: there should be only a tiny chance of ending up with less than you need. That's chance-constrained portfolio selection—and the tiny chance is only as good as your guesses about the future.

Investing With a Safety Rule

When people invest, they split money among different things like stocks, and the future value is uncertain. Chance-constrained portfolio selection means choosing that split to get the best expected result, but only among choices that keep the chance of falling below a 'safe' amount very small—for example, at most 5%. To use it, you need a model of how investments might go up and down and how they move together. The safety promise is only as trustworthy as that model: if the model is wrong, the real chance of a loss could be bigger. Things like trading costs and not knowing the true numbers also change which choices count as allowed.

Probability-Bounded Portfolio Choice

Chance-constrained portfolio selection is a way of choosing investment weights under uncertainty. The weights determine a random final wealth or loss at the end of a chosen time period. A chance constraint says the probability of falling below a survival threshold must be no more than a stated limit, and among portfolios that satisfy it, an optimizer picks the one with the best expected return or expected-utility objective. Solving it usually means turning the probability condition into an equivalent or approximate deterministic one, or using numerical methods. The guarantee depends on the assumed return distribution, time horizon, and how assets move together; if those are wrong, real failure rates can be higher. Loss aversion, value-at-risk-like constraints, transaction costs, and uncertainty in estimates all change which portfolios are feasible, so results should be checked out of sample.

Chance-constrained portfolio selection is a financial optimization framework in which asset weights, subject to budget and possibly short-sale constraints, determine a random terminal outcome such as final wealth or loss over a specified horizon. A chance constraint restricts the lower tail: the probability that the outcome falls below a survival threshold must not exceed a declared maximum violation probability. Among allocations satisfying this, an optimizer selects the one maximizing a declared criterion such as expected return or expected utility. Solutions typically rely on a deterministic reformulation, for instance under distributional assumptions, or on numerical and sampling-based solvers. The probability guarantee is only as reliable as the return distribution, horizon, and dependence model used, and the feasible set shifts with loss aversion, value-at-risk-type constraints, transaction costs, and estimation uncertainty. A complete specification therefore makes explicit the asset universe and horizon, weight constraints, return and dependence model, outcome expression, threshold, violation probability, objective, solution method, and robustness and out-of-sample validation.

Structural Signature¶

Carrier: the typed financial optimization carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets
Inputs or antecedent state: the exact financial optimization carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Chance-constrained portfolio selection
Constitutive operation: Asset weights determine a random terminal outcome, a chance constraint restricts its lower-tail failure probability and an optimizer selects a feasible allocation with the best declared expected utility or return criterion.
Invariant: the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit
Recognition test: type the carrier, state every parameter and convention in the definition, test that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases
Output or consequence: recognizing and comparing instances of Chance-constrained portfolio selection, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions
Failure boundary: the carrier is mistyped, the condition that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test

What It Is Not¶

It is not the whole field of financial optimization. The field contains many questions and methods that do not instantiate Chance-constrained portfolio selection.
It is not its most familiar example. A canonical instance directly demonstrates that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit. exhibits the structure, but the example is evidence for the abstraction rather than its definition.
It is not the neighboring catalog concept Mean–variance optimization. Mean–variance controls expected return and variance; chance-constrained selection directly bounds the modeled probability of breaching a threshold.
It is not a claim that every boundary case has one uncontested classification. a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Chance-constrained portfolio selection must control the decision
It is not an unrestricted metaphor for any process that seems similar. Outside financial optimization, the vocabulary and validity conditions do not transfer literally.

Scope of Application¶

Chance-constrained portfolio selection belongs to financial optimization and is useful where the analyst can specify the typed financial optimization carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit. The scope is broad within that domain but bounded by the need for the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit. Descriptive financial mathematics only; not investment advice or a recommendation to use any allocation.^[2]

Definition and recognition. Determine whether a proposed instance satisfies the constitutive conditions rather than merely sharing terminology.
Construction or evolution. Track how the exact financial optimization carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Chance-constrained portfolio selection are converted, constrained, or organized by Asset weights determine a random terminal outcome, a chance constraint restricts its lower-tail failure probability and an optimizer selects a feasible allocation with the best declared expected utility or return criterion..
Comparison. Compare instances using carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior, without treating convenience measures as the definition.
Boundary analysis. Diagnose cases where a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Chance-constrained portfolio selection must control the decision and state which convention or theorem controls the decision.
Downstream reasoning. Use the established identity to support recognizing and comparing instances of Chance-constrained portfolio selection, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions while preserving the assumptions under which the inference is valid.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Chance-constrained portfolio selection can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated. The disciplined statement is: given the exact financial optimization carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Chance-constrained portfolio selection, the structure counts as Chance-constrained portfolio selection exactly when the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit.

This format also separates identity from measurement. Empirical, computational, or documentary proxies support recognition only under declared validity and uncertainty assumptions; formal cases require proof rather than measurement. Measurements can be noisy, implementations can approximate, and proofs can use equivalent characterizations; none of those facts licenses changing the object being measured. When reports disagree, first check scope and convention, then data or proof, and only then interpret the disagreement as substantive.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Chance-constrained portfolio selection. Chance-constrained portfolio selection compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

The compression has a price. A single label can hide canonical, generalized, restricted, approximate, computational, empirical, and historically variant formulations of Chance-constrained portfolio selection. Good use therefore carries a small declaration of assumptions alongside the name. The abstraction manages complexity when it reduces the state space of the question while keeping the failure boundary visible; it mismanages complexity when the label substitutes for that boundary analysis.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed financial optimization carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit independently of one notation or implementation. This step prevents the canonical example from becoming the definition.
Derive consequences. From the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit, infer recognizing and comparing instances of Chance-constrained portfolio selection, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Record each assumption used so that a later change of setting does not silently preserve an invalid conclusion.
Test adversarial cases. Examine a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Chance-constrained portfolio selection must control the decision and an object that resembles Chance-constrained portfolio selection in purpose or vocabulary but does not satisfy its invariant is outside the class. A robust identity explains why the first is convention-sensitive and why the second is outside the class.
Compare and refine. Use carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior to compare legitimate instances, and refine the model when discrepancies reflect hidden variation rather than failure of the abstraction itself.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of financial optimization because they reuse the typed financial optimization carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Asset weights determine a random terminal outcome, a chance constraint restricts its lower-tail failure probability and an optimizer selects a feasible allocation with the best declared expected utility or return criterion., and type the carrier, state every parameter and convention in the definition, test that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases. A theorem, diagnostic, or modeling warning can travel when those roles remain literal. For example, the distinction between constitutive identity and a convenient observable transfers from A canonical instance directly demonstrates that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..^[3]

Transfer outside the home domain is weaker. The skeletal pattern—type the carrier, apply the defining mechanism of Chance-constrained portfolio selection, preserve its invariant, and derive only consequences licensed by the stated boundary—may suggest an analogy, but the domain-specific mechanisms, admissible evidence, and consequences do not come along automatically. The safe transfer procedure maps each role explicitly, checks the invariant again, and refuses the name when only a superficial resemblance remains.

Examples¶

Canonical¶

A canonical instance directly demonstrates that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit. The example exposes the carrier and directly tests that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit; changing incidental notation preserves the identity, while removing that condition destroys it. This example is canonical because every role can be inspected: the carrier is the typed financial optimization carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets; the operative rule is Asset weights determine a random terminal outcome, a chance constraint restricts its lower-tail failure probability and an optimizer selects a feasible allocation with the best declared expected utility or return criterion.; the invariant is the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit; and the result supports recognizing and comparing instances of Chance-constrained portfolio selection, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions.^[1] Changing incidental notation or scale leaves the structure intact, while removing the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit destroys the classification.

Mapped back: the typed financial optimization carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets → Asset weights determine a random terminal outcome, a chance constraint restricts its lower-tail failure probability and an optimizer selects a feasible allocation with the best declared expected utility or return criterion. → the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit → recognizing and comparing instances of Chance-constrained portfolio selection, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions

Applied / In Practice¶

An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation. The applied case qualifies only because the same invariant and boundary test remain literal under changed parameters or implementation. The applied case is not licensed merely by vocabulary. It qualifies because the same recognition test—type the carrier, state every parameter and convention in the definition, test that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases—can be run and because the same failure boundary—the carrier is mistyped, the condition that the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test—remains meaningful.^[2] The case also shows why practical outputs should report assumptions, resolution, and uncertainty instead of a naked label.

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Axiomatic identity vs. operational recognition. The defining conditions may be exact while empirical or computational recognition is approximate. Neither pole can be removed without changing the analytical task. Diagnostic: Can the reviewer state both the exact condition and the evidence used to infer it?
T2: Local roles vs. global consequence. The mechanism is enacted through local relations, but the abstraction is usually valued for a global classification or prediction. Neither pole can be removed without changing the analytical task. Diagnostic: Does the claimed global result actually follow from the declared local conditions?
T3: Ideal form vs. finite representation. Theory states a clean invariant while data structures, measurements, or proofs expose only finite representations. Neither pole can be removed without changing the analytical task. Diagnostic: Would increasing resolution converge toward the same classification?
T4: Canonical convention vs. legitimate variants. A standard formulation supports communication, while variants may preserve the same core under changed assumptions. Neither pole can be removed without changing the analytical task. Diagnostic: Which role is invariant across variants, and which convention-specific conclusion changes?
T5: Compression vs. hidden assumptions. The name compresses a complex argument but can conceal prerequisites. Neither pole can be removed without changing the analytical task. Diagnostic: Can each downstream inference be traced to an explicit assumption?
T6: Autonomous residual vs. reduction to catalog neighbors. The candidate uses broader structures but adds an identity-bearing residual. Neither pole can be removed without changing the analytical task. Diagnostic: After subtracting the proposed parent and named neighbors, does the constitutive residual still support independent diagnostics?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is type the carrier, apply the defining mechanism of Chance-constrained portfolio selection, preserve its invariant, and derive only consequences licensed by the stated boundary. Its identity-bearing terms—Chance-constrained portfolio selection, carrier, parameter, invariant, boundary, evidence, model, transformation, and application—derive their meaning from financial optimization and cannot be replaced by generic systems language without losing the tests that distinguish valid from invalid instances.

This mixed character explains why the abstraction is reusable inside the domain yet does not meet the Prime bar. The structure organizes reasoning, but its claims still depend on domain-specific objects, evidence, and intervention semantics.

Structural Core vs. Domain Accent¶

The structural core consists of a carrier, Asset weights determine a random terminal outcome, a chance constraint restricts its lower-tail failure probability and an optimizer selects a feasible allocation with the best declared expected utility or return criterion., a recognition invariant, and a consequence. That skeleton may resemble patterns elsewhere, especially type the carrier, apply the defining mechanism of Chance-constrained portfolio selection, preserve its invariant, and derive only consequences licensed by the stated boundary. The domain accent is not decorative: Chance-constrained portfolio selection, carrier, parameter, invariant, boundary, evidence, model, transformation, and application determine what counts as an admissible carrier, a valid transition, and successful evidence.

The abstraction therefore remains domain-specific. A cross-domain reuse that preserves only words such as 'balance,' 'cut,' 'sequence,' 'loss,' or 'simulation' is metaphor. Literal transfer requires the original role structure and diagnostics, which in this case remain anchored in financial optimization.

This entry is a kind of Constraint.

It remains its own entry because its identity is fixed by the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation.

Relationships to Other Abstractions¶

Current abstraction Chance-constrained portfolio selection Domain-specific

Parents (1) — outgoing typed relations

Chance-constrained portfolio selection is a kind of Constraint Prime

It remains its own entry because its identity is fixed by the asset universe and horizon, portfolio weights and budget and short-sale constraints, random return model and dependence, final wealth or loss expression, survival threshold, maximum violation probability, objective function, deterministic reformulation or numerical solver and estimation robustness and out-of-sample validation.

Hierarchy path (1) — routes to 1 parentless root

Chance-constrained portfolio selection → is a kind of Constraint

Neighborhood in Abstraction Space¶

Chance-constrained portfolio selection sits in a crowded region of the domain-specific corpus (33^rd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Optimization Algorithms & Methods (19 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With¶

Mean–variance optimization. Mean–variance controls expected return and variance; chance-constrained selection directly bounds the modeled probability of breaching a threshold.
One canonical example. An instance demonstrates the structure but does not define the whole abstraction.
Measurement or implementation of Chance-constrained portfolio selection. A proxy or realization is evidence for the abstraction, not the abstraction itself.
Generalized Chance-constrained portfolio selection. An extension qualifies only when its changed axioms and retained invariant are stated.

References¶

[1] A. Chance and W. W. Cooper (1959), "Chance-Constrained Programming," Management Science, 6, No. 1, 73-79. [https://www.jstor.org/stable/2627476?Search=yes&resultItemClick=true&searchText=((Management%20Science)%20AND%20(Charnes%20and%20cooper))%20AND%20(Chance%20Constrained%20programming)&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3DChance%2BConstrained%2Bprogramming%26cty_journal_facet%3Dam91cm5hbA%253D%253D%26prq%3D(Management%2BScience)%2BAND%2B(Charnes%2Band%2Bcooper)%26swp%3Don&ab_segments=0%2Fbasic_search_SYC-5455%2Fcontrol&refreqid=fastly-default%3A78e869b443ea6c0e104737db22b5a854]. Retrieved September 24, 2020. registry ↩a ↩b

[2] Naslund, B. and A. Whinston (1962), "A Model of Multi-Period Investment under Uncertainty," Management Science, 8, No. 2, 184-200. [https://www.jstor.org/stable/2627501?Search=yes&resultItemClick=true&searchText=(management%20science)%20AND%20(Naslund%20Bertil)&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3DNaslund%2BBertil%26prq%3Dmanagement%2Bscience%26swp%3Don&ab_segments=0%2Fbasic_search_SYC-5455%2Fcontrol&refreqid=fastly-default%3A99711b4afe59bd073fa1d58b5788eec2] Retrieved September 24, 2020. registry ↩a ↩b

[3] Agnew, N.H, R.A. Agnes, J. Rasmussen and K. R. Smith (1969), "An Application of Chance-Constrained Programming to Portfolio Selection in a Casualty Insurance Firm," Management Science, 15, No. 10, 512-520. [https://www.jstor.org/stable/2628459?Search=yes&resultItemClick=true&searchText=an%20application%20of%20chance%20constrained%20programming&searchUri=%2Faction%2FdoBasicSearch%3FQuery%3Dan%2Bapplication%2Bof%2Bchance%2Bconstrained%2Bprogramming&ab_segments=0%2Fbasic_search_SYC-5455%2Fcontrol&refreqid=fastly-default%3Ac66a912b771798f92fc98226d695291d]. Retrieved September 24, 2020. registry ↩