A Study of MLN9708 in Japanese Participants With Relapsed and/or Refractory Multiple Myeloma (RRMM)

Inclusion Criteria: 1. Japanese participants with multiple myeloma according to diagnostic criteria. 2. Previously treated with 2 or more regimens including all the following drugs; bortezomib, thalidomide or lenalidomide, corticosteroids. 3. Who have relapsed following the previous therapy or failed to continue the treatment due to their intolerability to the last treatment regimen for myeloma. 4. Measurable disease defined by at least one of the following 3 measurements; Serum M-protein: greater than or equal to (\>=) 1 gram per deciliter (g/dL) (\>= 10 gram per liter \[g/L\]), Urine M-protein: \>= 200 mg/24 hours, Serum free light chain (FLC) assay: involved FLC level \>= 10 milligram per deciliter (mg/dL) (\>= 100 milligram per liter \[mg/L\]), provided that the serum FLC ratio is abnormal. 5. Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2. 6. 20 years or older at giving their informed consent. 7. Must be able to stay in the hospital for Cycle 1 treatment. 8. Must meet the following laboratory criteria at screening; Absolute neutrophil count (ANC): \>=1,000 per cubic millimeter (/mm\^3), Platelet count: \>=75,000/mm\^3, Total bilirubin: \<=1.5\* the upper limit of normal range (ULN), Alanine aminotransferase (ALT) and aspartate aminotransferase (AST): less than or equal to (\<=) 3\* ULN, Creatinine clearance: calculated by using Cockcroft-Gault formula; MLN9708 monotherapy cohort: \>=30 milliliter per minute (mL/min); MLN9708 with Rd cohort: \>=60 mL/min. 9. Recovered (\<= Grade 1) from the toxicities of the prior treatments. ANC \>=1,000/mm\^3. 10. Life expectancy of at least 3 months, in the judgment of the investigator. 11. Conforming to proper management guidelines of lenalidomide (MLN9708 with Rd cohort only). Exclusion Criteria: 1. With plasmacytoma only. 2. With plasma cell leukemia. 3. With central nervous system invasion. 4. Radiotherapy within 14 days before enrollment. 5. Other anti-tumor drug administration within 21 days before enrollment. 6. Other investigational products administration within 21 days before enrollment (60 days from the last dose for carfilzomib). 7. Antibody treatment within 42 days before enrollment. 8. Systemic treatment with potent cytochrome P450 (CYP) isozyme 1A2 inhibitors (fluvoxamine, enoxacin), potent CYP3A inhibitors (clarithromycin, telithromycin, itraconazole, voriconazole, ketoconazole), or potent CYP3A inducers (rifampin, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of foods containing Ginkgo biloba extract, St. John's Wort, or grapefruit within 14 days before enrollment. 9. Treatment with corticosteroids greater than (\>) 10 mg of prednisolone per day. Inhaled and topical steroids are permitted. 10. Peripheral neuropathy \>=Grade 2. 11. Diarrhea \>= Grade 2. 12. Major surgery requiring general anesthesia within 14 days before enrollment. 13. Infection requiring systemic antibiotic treatment or other serious infections within 14 days before enrollment. 14. Evidence of concurrent uncontrolled cardiovascular conditions including hypertension, cardiac arrhythmias, New York Heart Association (NYHA) Class III or worse congestive heart failure, angina, myocardial infarction, or cerebral infarction within 6 months before enrollment. 15. Corrected QT interval (QTc) \> 470 milliseconds on a 12-lead ECG obtained during the screening period. 16. Tested positive for human immunodeficiency virus (HIV) antibody, hepatitis B virus surface antigen (HBs antigen), or hepatitis C virus (HCV) antibody during the screening period. 17. Hypersensitivity to MLN9708 (including excipients), boron, or boron-containing drugs. 18. Hypersensitivity to lenalidomide, or dexamethasone, or excipients contained in the formulation of each drug (MLN9708 with Rd cohort only). 19. Known gastrointestinal diseases (difficulty swallowing, inflamed gastroenteritis, and Crohn disease), or gastrointestinal procedure (endoscopic procedure is permitted), that could interfere with the oral absorption or tolerance of the study treatment. 20. Uncontrolled diabetes mellitus. 21. A history of interstitial lung disease or lung fibrosis, or a current complication of interstitial lung disease or lung fibrosis diagnosed by diagnostic chest imaging. 22. Prior or current complications of deep vein thrombosis or pulmonary embolism (MLN9708 with Rd cohort only). 23 Diagnosed or treated for another malignancy within 2 years before the first dose or previously diagnosed with another malignancy and have any evidence of residual disease. Participants with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have complete resection. 24\. Who do not consent to use adequate contraceptive precautions (example, condoms and oral contraceptives) during the following term: * For women with childbearing potential, from when giving their consent through 3 months after the last dose of MLN9708, dexamethasone, or lenalidomide * For men having their partners with childbearing potential, from giving their consent through 4 months after last dose of MLN9708, dexamethasone, or lenalidomide. 25\. Pregnant (example, positive for pregnancy test) or lactating. Lactation is prohibited from the first dose through 6 months after the last dose of MLN9708, dexamethasone, and lenalidomide. 26\. Use of an investigational medical device within 28 days before enrollment. 27. Any inabilities that could potentially interfere with the consent or completion of treatment according to this protocol. 28\. Having difficulties in participation to this study by the investigator's judgment.